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Theorem preqr1 3539
 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
preqr1.1 𝐴 ∈ V
preqr1.2 𝐵 ∈ V
Assertion
Ref Expression
preqr1 ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)

Proof of Theorem preqr1
StepHypRef Expression
1 preqr1.1 . . . . 5 𝐴 ∈ V
21prid1 3476 . . . 4 𝐴 ∈ {𝐴, 𝐶}
3 eleq2 2101 . . . 4 ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 ∈ {𝐴, 𝐶} ↔ 𝐴 ∈ {𝐵, 𝐶}))
42, 3mpbii 136 . . 3 ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 ∈ {𝐵, 𝐶})
51elpr 3396 . . 3 (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))
64, 5sylib 127 . 2 ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
7 preqr1.2 . . . . 5 𝐵 ∈ V
87prid1 3476 . . . 4 𝐵 ∈ {𝐵, 𝐶}
9 eleq2 2101 . . . 4 ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 ∈ {𝐴, 𝐶} ↔ 𝐵 ∈ {𝐵, 𝐶}))
108, 9mpbiri 157 . . 3 ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐵 ∈ {𝐴, 𝐶})
117elpr 3396 . . 3 (𝐵 ∈ {𝐴, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐶))
1210, 11sylib 127 . 2 ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 = 𝐴𝐵 = 𝐶))
13 eqcom 2042 . 2 (𝐴 = 𝐵𝐵 = 𝐴)
14 eqeq2 2049 . 2 (𝐴 = 𝐶 → (𝐵 = 𝐴𝐵 = 𝐶))
156, 12, 13, 14oplem1 882 1 ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 629   = wceq 1243   ∈ wcel 1393  Vcvv 2557  {cpr 3376 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382 This theorem is referenced by:  preqr2  3540
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