Theorem infinfnum

Description: Equivalence between two infiniteness criteria for numerable sets. (Contributed by BTernaryTau, 15-Jul-2026)

	Ref	Expression
Assertion	infinfnum	|- ( A e. dom card -> ( -. A e. Fin <-> _om ~<_ A ) )

Proof

Step	Hyp	Ref	Expression
1		isfin4-2	|- ( A e. dom card -> ( A e. Fin4 <-> -. _om ~<_ A ) )
2	1	con2bid	|- ( A e. dom card -> ( _om ~<_ A <-> -. A e. Fin4 ) )
3		fin45	|- ( A e. Fin4 -> A e. Fin5 )
4		fin56	|- ( A e. Fin5 -> A e. Fin6 )
5		fin67	|- ( A e. Fin6 -> A e. Fin7 )
6	3 4 5	3syl	|- ( A e. Fin4 -> A e. Fin7 )
7		fin71num	|- ( A e. dom card -> ( A e. Fin7 <-> A e. Fin ) )
8	6 7	imbitrid	|- ( A e. dom card -> ( A e. Fin4 -> A e. Fin ) )
9		fin12	|- ( A e. Fin -> A e. Fin2 )
10		fin23	|- ( A e. Fin2 -> A e. Fin3 )
11		fin34	|- ( A e. Fin3 -> A e. Fin4 )
12	9 10 11	3syl	|- ( A e. Fin -> A e. Fin4 )
13	8 12	impbid1	|- ( A e. dom card -> ( A e. Fin4 <-> A e. Fin ) )
14	13	notbid	|- ( A e. dom card -> ( -. A e. Fin4 <-> -. A e. Fin ) )
15	2 14	bitr2d	|- ( A e. dom card -> ( -. A e. Fin <-> _om ~<_ A ) )