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 e. _V
Assertion preleq
|- ( ( ( A e. B /\ C e. D ) /\ { A , B } = { C , D } ) -> ( A = C /\ B = D ) )

Proof

Step Hyp Ref Expression
1 preleq.b
 |-  B e. _V
2 preleqg
 |-  ( ( ( A e. B /\ B e. _V /\ C e. D ) /\ { A , B } = { C , D } ) -> ( A = C /\ B = D ) )
3 1 2 mp3anl2
 |-  ( ( ( A e. B /\ C e. D ) /\ { A , B } = { C , D } ) -> ( A = C /\ B = D ) )