Theorem elxnn0

Description: An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020)

	Ref	Expression
Assertion	elxnn0	|- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )

Proof

Step	Hyp	Ref	Expression
1		df-xnn0	|- NN0* = ( NN0 u. { +oo } )
2	1	eleq2i	|- ( A e. NN0* <-> A e. ( NN0 u. { +oo } ) )
3		elun	|- ( A e. ( NN0 u. { +oo } ) <-> ( A e. NN0 \/ A e. { +oo } ) )
4		pnfex	|- +oo e. _V
5	4	elsn2	|- ( A e. { +oo } <-> A = +oo )
6	5	orbi2i	|- ( ( A e. NN0 \/ A e. { +oo } ) <-> ( A e. NN0 \/ A = +oo ) )
7	2 3 6	3bitri	|- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )