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	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( card ‘ 𝐴 ) ⊆ ( card ‘ 𝐵 ) ↔ 𝐴 ≼ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		numth3	⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ dom card )
2		numth3	⊢ ( 𝐵 ∈ 𝑊 → 𝐵 ∈ dom card )
3		carddom2	⊢ ( ( 𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ) → ( ( card ‘ 𝐴 ) ⊆ ( card ‘ 𝐵 ) ↔ 𝐴 ≼ 𝐵 ) )
4	1 2 3	syl2an	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( card ‘ 𝐴 ) ⊆ ( card ‘ 𝐵 ) ↔ 𝐴 ≼ 𝐵 ) )