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Theorem preqr1OLD 4371
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.) Obsolete version of preqr1 4370 as of 18-Dec-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
preqr1.a 𝐴 ∈ V
preqr1.b 𝐵 ∈ V
Assertion
Ref Expression
preqr1OLD ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)

Proof of Theorem preqr1OLD
StepHypRef Expression
1 preqr1.a . . . . 5 𝐴 ∈ V
21prid1 4288 . . . 4 𝐴 ∈ {𝐴, 𝐶}
3 eleq2 2688 . . . 4 ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 ∈ {𝐴, 𝐶} ↔ 𝐴 ∈ {𝐵, 𝐶}))
42, 3mpbii 223 . . 3 ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 ∈ {𝐵, 𝐶})
51elpr 4189 . . 3 (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))
64, 5sylib 208 . 2 ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
7 preqr1.b . . . . 5 𝐵 ∈ V
87prid1 4288 . . . 4 𝐵 ∈ {𝐵, 𝐶}
9 eleq2 2688 . . . 4 ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 ∈ {𝐴, 𝐶} ↔ 𝐵 ∈ {𝐵, 𝐶}))
108, 9mpbiri 248 . . 3 ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐵 ∈ {𝐴, 𝐶})
117elpr 4189 . . 3 (𝐵 ∈ {𝐴, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐶))
1210, 11sylib 208 . 2 ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 = 𝐴𝐵 = 𝐶))
13 eqcom 2627 . 2 (𝐴 = 𝐵𝐵 = 𝐴)
14 eqeq2 2631 . 2 (𝐴 = 𝐶 → (𝐵 = 𝐴𝐵 = 𝐶))
156, 12, 13, 14oplem1 1006 1 ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383   = wceq 1481  wcel 1988  Vcvv 3195  {cpr 4170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-un 3572  df-sn 4169  df-pr 4171
This theorem is referenced by: (None)
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