Theorem gchacg

Description: A "local" form of gchac . If A and ~P A are GCH-sets, then ~P A is well-orderable. The proof is due to Specker. Theorem 2.1 of KanamoriPincus p. 419. (Contributed by Mario Carneiro, 15-May-2015)

	Ref	Expression
Assertion	gchacg	|- ( ( _om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A e. dom card )

Proof

Step	Hyp	Ref	Expression
1		harcl	|- ( har ` A ) e. On
2		gchhar	|- ( ( _om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( har ` A ) ~~ ~P A )
3		isnumi	|- ( ( ( har ` A ) e. On /\ ( har ` A ) ~~ ~P A ) -> ~P A e. dom card )
4	1 2 3	sylancr	|- ( ( _om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A e. dom card )