Theorem preqr1 4804

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.)

Hypotheses

Ref	Expression
preqr1.a	⊢ 𝐴 ∈ V
preqr1.b	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
preqr1	⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)

Proof of Theorem preqr1

Step	Hyp	Ref	Expression
1		preqr1.a	. . 3 ⊢ 𝐴 ∈ V
2		id 22	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ V)
3		preqr1.b	. . . . 5 ⊢ 𝐵 ∈ V
4	3	a1i 11	. . . 4 ⊢ (𝐴 ∈ V → 𝐵 ∈ V)
5	2, 4	preq1b 4802	. . 3 ⊢ (𝐴 ∈ V → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
6	1, 5	ax-mp 5	. 2 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵)
7	6	biimpi 216	1 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113  Vcvv 3440  {cpr 4582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-un 3906  df-sn 4581  df-pr 4583

This theorem is referenced by:  preqr2  4805  opthwiener  5462  opthhausdorff0  5466  cusgrfilem2  29530  usgr2wlkneq  29829  wopprc  43282