Theorem fineqvacALT

Description: Shorter proof of fineqvac using ax-rep and ax-pow . (Contributed by BTernaryTau, 21-Sep-2024) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	fineqvacALT	⊢ ( Fin = V → CHOICE )

Proof

Step	Hyp	Ref	Expression
1		ssv	⊢ dom card ⊆ V
2	1	a1i	⊢ ( Fin = V → dom card ⊆ V )
3		finnum	⊢ ( 𝑥 ∈ Fin → 𝑥 ∈ dom card )
4	3	ssriv	⊢ Fin ⊆ dom card
5		sseq1	⊢ ( Fin = V → ( Fin ⊆ dom card ↔ V ⊆ dom card ) )
6	4 5	mpbii	⊢ ( Fin = V → V ⊆ dom card )
7	2 6	eqssd	⊢ ( Fin = V → dom card = V )
8		dfac10	⊢ ( CHOICE ↔ dom card = V )
9	7 8	sylibr	⊢ ( Fin = V → CHOICE )