Theorem preqr1 4853

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 4851	. . 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 1537   ∈ wcel 2106  Vcvv 3478  {cpr 4633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634

This theorem is referenced by:  preqr2  4854  opthwiener  5524  opthhausdorff0  5528  cusgrfilem2  29489  usgr2wlkneq  29789  wopprc  43019