"I never doubted the character of the players", said O'Neill.
"And there are about six others I'd rather not have, but I am delighted to be in it".
Should Ireland navigate the play-offs and reach the tournament it will be the fourth time in their history they have qualified for a World Cup.
A 1-0 win saw Ireland leapfrog opponents Wales into second place in Group D and clinch a place in November's play-offs.
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As for their conquerors, a gracious Coleman wished the Irish well for the next stage of their World Cup quest, saying: "Martin O'Neill has done a great job and good luck to them".
"I think our approach will be similar to the one that we had against Italy in Lille where we knew before the game that a win was the only thing that mattered for us".
All will become clear who the Republic will take on when the play-off draw for Russian Federation is made next Tuesday, but whoever they face, Meyler is confident Martin O'Neil's side will be joining England at the tournament next summer. But our players have done brilliantly and we've got to learn from it'. And then we go out and beat them.
He has 26 goals in 68 worldwide appearances, including key strikes against Georgia and Serbia in the current campaign which should have put Wales on the road to Russian Federation.
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Asked if his misfortune had provided added motivation, Arter said: "Yes, but that aside, even if I did go to the Euros, to play in a World Cup is".
"We've got a lot of work to do".
IT wasn't to be for Joe Allen and Wales last night as their World Cup dream died at the hands of the Republic of Ireland in Cardiff. To come and win in Wales was fantastic and the players will not give up - that's the point. "Why you'd want him to leave I'm not too sure".
"His energy - when you see James getting exhausted in a game with 10 minutes to go, you realise the rest of the players must be really exhausted because James just keeps going".
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The former West Ham, Liverpool, Chelsea, Arsenal and QPR midfielder is now seeing out his days at Beitar Jerusalem. Spain's Asier Illarramendi scored a sweet strike against Israel on Monday as he grabbed his first worldwide goal.
Not that we want to jump ahead of ourselves (we so want to jump ahead of ourselves, for goodness sake we're checking out flights to Russian Federation as we type!) but the draw for the 2018 World Cup will take place on Friday, December 1st in Moscow.