Theorem preqr1

Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. (Contributed by NM, 18-Oct-1995)

	Ref	Expression
Hypotheses	preqr1.a	⊢ 𝐴 ∈ V
	preqr1.b	⊢ 𝐵 ∈ V
Assertion	preqr1	⊢ ( { 𝐴 , 𝐶 } = { 𝐵 , 𝐶 } → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		preqr1.a	⊢ 𝐴 ∈ V
2		preqr1.b	⊢ 𝐵 ∈ V
3		id	⊢ ( 𝐴 ∈ V → 𝐴 ∈ V )
4	2	a1i	⊢ ( 𝐴 ∈ V → 𝐵 ∈ V )
5	3 4	preq1b	⊢ ( 𝐴 ∈ V → ( { 𝐴 , 𝐶 } = { 𝐵 , 𝐶 } ↔ 𝐴 = 𝐵 ) )
6	1 5	ax-mp	⊢ ( { 𝐴 , 𝐶 } = { 𝐵 , 𝐶 } ↔ 𝐴 = 𝐵 )
7	6	biimpi	⊢ ( { 𝐴 , 𝐶 } = { 𝐵 , 𝐶 } → 𝐴 = 𝐵 )