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Theorem preqr1 3634
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 3568 . . . 4 𝐴 ∈ {𝐴, 𝐶}
3 eleq2 2158 . . . 4 ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 ∈ {𝐴, 𝐶} ↔ 𝐴 ∈ {𝐵, 𝐶}))
42, 3mpbii 147 . . 3 ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 ∈ {𝐵, 𝐶})
51elpr 3487 . . 3 (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))
64, 5sylib 121 . 2 ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
7 preqr1.2 . . . . 5 𝐵 ∈ V
87prid1 3568 . . . 4 𝐵 ∈ {𝐵, 𝐶}
9 eleq2 2158 . . . 4 ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 ∈ {𝐴, 𝐶} ↔ 𝐵 ∈ {𝐵, 𝐶}))
108, 9mpbiri 167 . . 3 ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐵 ∈ {𝐴, 𝐶})
117elpr 3487 . . 3 (𝐵 ∈ {𝐴, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐶))
1210, 11sylib 121 . 2 ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 = 𝐴𝐵 = 𝐶))
13 eqcom 2097 . 2 (𝐴 = 𝐵𝐵 = 𝐴)
14 eqeq2 2104 . 2 (𝐴 = 𝐶 → (𝐵 = 𝐴𝐵 = 𝐶))
156, 12, 13, 14oplem1 924 1 ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 667   = wceq 1296  wcel 1445  Vcvv 2633  {cpr 3467
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-un 3017  df-sn 3472  df-pr 3473
This theorem is referenced by:  preqr2  3635
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