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Theorem preqr1g 3511
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 3513. (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 3448 . . . . . . 7 (A V → A {A, 𝐶})
2 eleq2 2083 . . . . . . 7 ({A, 𝐶} = {B, 𝐶} → (A {A, 𝐶} ↔ A {B, 𝐶}))
31, 2syl5ibcom 144 . . . . . 6 (A V → ({A, 𝐶} = {B, 𝐶} → A {B, 𝐶}))
4 elprg 3367 . . . . . 6 (A V → (A {B, 𝐶} ↔ (A = B A = 𝐶)))
53, 4sylibd 138 . . . . 5 (A V → ({A, 𝐶} = {B, 𝐶} → (A = B A = 𝐶)))
65adantr 261 . . . 4 ((A V B V) → ({A, 𝐶} = {B, 𝐶} → (A = B A = 𝐶)))
76imp 115 . . 3 (((A V B V) {A, 𝐶} = {B, 𝐶}) → (A = B A = 𝐶))
8 prid1g 3448 . . . . . . 7 (B V → B {B, 𝐶})
9 eleq2 2083 . . . . . . 7 ({A, 𝐶} = {B, 𝐶} → (B {A, 𝐶} ↔ B {B, 𝐶}))
108, 9syl5ibrcom 146 . . . . . 6 (B V → ({A, 𝐶} = {B, 𝐶} → B {A, 𝐶}))
11 elprg 3367 . . . . . 6 (B V → (B {A, 𝐶} ↔ (B = A B = 𝐶)))
1210, 11sylibd 138 . . . . 5 (B V → ({A, 𝐶} = {B, 𝐶} → (B = A B = 𝐶)))
1312adantl 262 . . . 4 ((A V B V) → ({A, 𝐶} = {B, 𝐶} → (B = A B = 𝐶)))
1413imp 115 . . 3 (((A V B V) {A, 𝐶} = {B, 𝐶}) → (B = A B = 𝐶))
15 eqcom 2024 . . 3 (A = BB = A)
16 eqeq2 2031 . . 3 (A = 𝐶 → (B = AB = 𝐶))
177, 14, 15, 16oplem1 870 . 2 (((A V B V) {A, 𝐶} = {B, 𝐶}) → A = B)
1817ex 108 1 ((A V B V) → ({A, 𝐶} = {B, 𝐶} → A = B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 616   = wceq 1228   wcel 1374  Vcvv 2535  {cpr 3351
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-sn 3356  df-pr 3357
This theorem is referenced by:  preqr2g  3512
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