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Theorem preqr1g 3781
Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 3783. (Contributed by Jim Kingdon, 21-Sep-2018.)
Assertion
Ref Expression
preqr1g ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))

Proof of Theorem preqr1g
StepHypRef Expression
1 prid1g 3711 . . . . . . 7 (𝐴 ∈ V → 𝐴 ∈ {𝐴, 𝐶})
2 eleq2 2253 . . . . . . 7 ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 ∈ {𝐴, 𝐶} ↔ 𝐴 ∈ {𝐵, 𝐶}))
31, 2syl5ibcom 155 . . . . . 6 (𝐴 ∈ V → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 ∈ {𝐵, 𝐶}))
4 elprg 3627 . . . . . 6 (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
53, 4sylibd 149 . . . . 5 (𝐴 ∈ V → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶)))
65adantr 276 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶)))
76imp 124 . . 3 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → (𝐴 = 𝐵𝐴 = 𝐶))
8 prid1g 3711 . . . . . . 7 (𝐵 ∈ V → 𝐵 ∈ {𝐵, 𝐶})
9 eleq2 2253 . . . . . . 7 ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 ∈ {𝐴, 𝐶} ↔ 𝐵 ∈ {𝐵, 𝐶}))
108, 9syl5ibrcom 157 . . . . . 6 (𝐵 ∈ V → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐵 ∈ {𝐴, 𝐶}))
11 elprg 3627 . . . . . 6 (𝐵 ∈ V → (𝐵 ∈ {𝐴, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐶)))
1210, 11sylibd 149 . . . . 5 (𝐵 ∈ V → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 = 𝐴𝐵 = 𝐶)))
1312adantl 277 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 = 𝐴𝐵 = 𝐶)))
1413imp 124 . . 3 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → (𝐵 = 𝐴𝐵 = 𝐶))
15 eqcom 2191 . . 3 (𝐴 = 𝐵𝐵 = 𝐴)
16 eqeq2 2199 . . 3 (𝐴 = 𝐶 → (𝐵 = 𝐴𝐵 = 𝐶))
177, 14, 15, 16oplem1 977 . 2 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → 𝐴 = 𝐵)
1817ex 115 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 709   = wceq 1364  wcel 2160  Vcvv 2752  {cpr 3608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614
This theorem is referenced by:  preqr2g  3782
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