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 ( ( ( 𝐴𝐵𝐶𝐷 ) ∧ { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } ) → ( 𝐴 = 𝐶𝐵 = 𝐷 ) )

Proof

Step Hyp Ref Expression
1 preleq.b 𝐵 ∈ V
2 preleqg ( ( ( 𝐴𝐵𝐵 ∈ V ∧ 𝐶𝐷 ) ∧ { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } ) → ( 𝐴 = 𝐶𝐵 = 𝐷 ) )
3 1 2 mp3anl2 ( ( ( 𝐴𝐵𝐶𝐷 ) ∧ { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } ) → ( 𝐴 = 𝐶𝐵 = 𝐷 ) )