Metamath Proof Explorer

Theorem preleq

Description: Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996) (Revised by AV, 15-Jun-2022)

Ref Expression
Hypothesis preleq.b ⊢ B ∈ V
Assertion preleq ⊢ A ∈ B ∧ C ∈ D ∧ A B = C D → A = C ∧ B = D

Proof

Step Hyp Ref Expression
1 preleq.b ⊢ B ∈ V
2 preleqg ⊢ A ∈ B ∧ B ∈ V ∧ C ∈ D ∧ A B = C D → A = C ∧ B = D
3 1 2 mp3anl2 ⊢ A ∈ B ∧ C ∈ D ∧ A B = C D → A = C ∧ B = D