Theorem cardidg

Description: Any set is equinumerous to its cardinal number. Closed theorem form of cardid . (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Assertion	cardidg	$$⊢A\in B\to \mathrm{card}\left(A\right)\approx A$$

Proof

Step	Hyp	Ref	Expression
1		elex	$$⊢A\in B\to A\in \mathrm{V}$$
2		cardeqv	$$⊢\mathrm{dom}\mathrm{card}=\mathrm{V}$$
3	2	eleq2i	$$⊢A\in \mathrm{dom}\mathrm{card}\leftrightarrow A\in \mathrm{V}$$
4		cardid2	$$⊢A\in \mathrm{dom}\mathrm{card}\to \mathrm{card}\left(A\right)\approx A$$
5	3 4	sylbir	$$⊢A\in \mathrm{V}\to \mathrm{card}\left(A\right)\approx A$$
6	1 5	syl	$$⊢A\in B\to \mathrm{card}\left(A\right)\approx A$$