Theorem cardid

Description: Any set is equinumerous to its cardinal number. Proposition 10.5 of TakeutiZaring p. 85. (Contributed by NM, 22-Oct-2003) (Revised by Mario Carneiro, 28-Apr-2015)

	Ref	Expression
Hypothesis	cardval.1	$$⊢A\in \mathrm{V}$$
Assertion	cardid	$$⊢\mathrm{card}\left(A\right)\approx A$$

Proof

Step	Hyp	Ref	Expression
1		cardval.1	$$⊢A\in \mathrm{V}$$
2		numth3	$$⊢A\in \mathrm{V}\to A\in \mathrm{dom}\mathrm{card}$$
3		cardid2	$$⊢A\in \mathrm{dom}\mathrm{card}\to \mathrm{card}\left(A\right)\approx A$$
4	1 2 3	mp2b	$$⊢\mathrm{card}\left(A\right)\approx A$$