Theorem carddom

Description: Two sets have the dominance relationship iff their cardinalities have the subset relationship. Equation i of Quine p. 232. (Contributed by NM, 22-Oct-2003) (Revised by Mario Carneiro, 30-Apr-2015)

	Ref	Expression
Assertion	carddom	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\mathrm{card}\left(A\right)\subseteq \mathrm{card}\left(B\right)\leftrightarrow A\preccurlyeq B\right)$$

Proof

Step	Hyp	Ref	Expression
1		numth3	$$⊢A\in V\to A\in \mathrm{dom}\mathrm{card}$$
2		numth3	$$⊢B\in W\to B\in \mathrm{dom}\mathrm{card}$$
3		carddom2	$$⊢\left(A\in \mathrm{dom}\mathrm{card}\wedge B\in \mathrm{dom}\mathrm{card}\right)\to \left(\mathrm{card}\left(A\right)\subseteq \mathrm{card}\left(B\right)\leftrightarrow A\preccurlyeq B\right)$$
4	1 2 3	syl2an	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\mathrm{card}\left(A\right)\subseteq \mathrm{card}\left(B\right)\leftrightarrow A\preccurlyeq B\right)$$