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Theorem preqr1g 3632
 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 3634. (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 3566 . . . . . . 7 (𝐴 ∈ V → 𝐴 ∈ {𝐴, 𝐶})
2 eleq2 2158 . . . . . . 7 ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 ∈ {𝐴, 𝐶} ↔ 𝐴 ∈ {𝐵, 𝐶}))
31, 2syl5ibcom 154 . . . . . 6 (𝐴 ∈ V → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 ∈ {𝐵, 𝐶}))
4 elprg 3486 . . . . . 6 (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
53, 4sylibd 148 . . . . 5 (𝐴 ∈ V → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶)))
65adantr 271 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶)))
76imp 123 . . 3 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → (𝐴 = 𝐵𝐴 = 𝐶))
8 prid1g 3566 . . . . . . 7 (𝐵 ∈ V → 𝐵 ∈ {𝐵, 𝐶})
9 eleq2 2158 . . . . . . 7 ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 ∈ {𝐴, 𝐶} ↔ 𝐵 ∈ {𝐵, 𝐶}))
108, 9syl5ibrcom 156 . . . . . 6 (𝐵 ∈ V → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐵 ∈ {𝐴, 𝐶}))
11 elprg 3486 . . . . . 6 (𝐵 ∈ V → (𝐵 ∈ {𝐴, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐶)))
1210, 11sylibd 148 . . . . 5 (𝐵 ∈ V → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 = 𝐴𝐵 = 𝐶)))
1312adantl 272 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 = 𝐴𝐵 = 𝐶)))
1413imp 123 . . 3 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → (𝐵 = 𝐴𝐵 = 𝐶))
15 eqcom 2097 . . 3 (𝐴 = 𝐵𝐵 = 𝐴)
16 eqeq2 2104 . . 3 (𝐴 = 𝐶 → (𝐵 = 𝐴𝐵 = 𝐶))
177, 14, 15, 16oplem1 924 . 2 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → 𝐴 = 𝐵)
1817ex 114 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∨ 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:  preqr2g  3633
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