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 ) ) |
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 ) ) |