Theorem cardval

Description: The value of the cardinal number function. Definition 10.4 of TakeutiZaring p. 85. See cardval2 for a simpler version of its value. (Contributed by NM, 21-Oct-2003) (Revised by Mario Carneiro, 28-Apr-2015)

	Ref	Expression
Hypothesis	cardval.1	$$⊢A\in \mathrm{V}$$
Assertion	cardval	$$⊢\mathrm{card}\left(A\right)=\bigcap \left\{x\in \mathrm{On}|x\approx A\right\}$$

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		cardval3	$$⊢A\in \mathrm{dom}\mathrm{card}\to \mathrm{card}\left(A\right)=\bigcap \left\{x\in \mathrm{On}|x\approx A\right\}$$
4	1 2 3	mp2b	$$⊢\mathrm{card}\left(A\right)=\bigcap \left\{x\in \mathrm{On}|x\approx A\right\}$$