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	$$⊢\mathrm{Fin}=\mathrm{V}\to \mathrm{CHOICE}$$

Proof

Step	Hyp	Ref	Expression
1		ssv	$$⊢\mathrm{dom}\mathrm{card}\subseteq \mathrm{V}$$
2	1	a1i	$$⊢\mathrm{Fin}=\mathrm{V}\to \mathrm{dom}\mathrm{card}\subseteq \mathrm{V}$$
3		finnum	$$⊢x\in \mathrm{Fin}\to x\in \mathrm{dom}\mathrm{card}$$
4	3	ssriv	$$⊢\mathrm{Fin}\subseteq \mathrm{dom}\mathrm{card}$$
5		sseq1	$$⊢\mathrm{Fin}=\mathrm{V}\to \left(\mathrm{Fin}\subseteq \mathrm{dom}\mathrm{card}\leftrightarrow \mathrm{V}\subseteq \mathrm{dom}\mathrm{card}\right)$$
6	4 5	mpbii	$$⊢\mathrm{Fin}=\mathrm{V}\to \mathrm{V}\subseteq \mathrm{dom}\mathrm{card}$$
7	2 6	eqssd	$$⊢\mathrm{Fin}=\mathrm{V}\to \mathrm{dom}\mathrm{card}=\mathrm{V}$$
8		dfac10	$$⊢\mathrm{CHOICE}\leftrightarrow \mathrm{dom}\mathrm{card}=\mathrm{V}$$
9	7 8	sylibr	$$⊢\mathrm{Fin}=\mathrm{V}\to \mathrm{CHOICE}$$