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Theorem preqr1 3513
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 3450 . . . 4 A {A, 𝐶}
3 eleq2 2083 . . . 4 ({A, 𝐶} = {B, 𝐶} → (A {A, 𝐶} ↔ A {B, 𝐶}))
42, 3mpbii 136 . . 3 ({A, 𝐶} = {B, 𝐶} → A {B, 𝐶})
51elpr 3368 . . 3 (A {B, 𝐶} ↔ (A = B A = 𝐶))
64, 5sylib 127 . 2 ({A, 𝐶} = {B, 𝐶} → (A = B A = 𝐶))
7 preqr1.2 . . . . 5 B V
87prid1 3450 . . . 4 B {B, 𝐶}
9 eleq2 2083 . . . 4 ({A, 𝐶} = {B, 𝐶} → (B {A, 𝐶} ↔ B {B, 𝐶}))
108, 9mpbiri 157 . . 3 ({A, 𝐶} = {B, 𝐶} → B {A, 𝐶})
117elpr 3368 . . 3 (B {A, 𝐶} ↔ (B = A B = 𝐶))
1210, 11sylib 127 . 2 ({A, 𝐶} = {B, 𝐶} → (B = A B = 𝐶))
13 eqcom 2024 . 2 (A = BB = A)
14 eqeq2 2031 . 2 (A = 𝐶 → (B = AB = 𝐶))
156, 12, 13, 14oplem1 870 1 ({A, 𝐶} = {B, 𝐶} → A = B)
Colors of variables: wff set class
Syntax hints:  wi 4   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:  preqr2  3514
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