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Metamath Proof Explorer

Theorem cardsdom

Description: Two sets have the strict dominance relationship iff their cardinalities have the membership relationship. Corollary 19.7(2) of Eisenberg p. 310. (Contributed by NM, 22-Oct-2003) (Revised by Mario Carneiro, 30-Apr-2015)

Ref Expression
Assertion cardsdom ⊢ 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 cardsdom2 ⊢ 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