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Theorem preqr1 3530
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 3467 . . . 4 A {A, 𝐶}
3 eleq2 2098 . . . 4 ({A, 𝐶} = {B, 𝐶} → (A {A, 𝐶} ↔ A {B, 𝐶}))
42, 3mpbii 136 . . 3 ({A, 𝐶} = {B, 𝐶} → A {B, 𝐶})
51elpr 3385 . . 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 3467 . . . 4 B {B, 𝐶}
9 eleq2 2098 . . . 4 ({A, 𝐶} = {B, 𝐶} → (B {A, 𝐶} ↔ B {B, 𝐶}))
108, 9mpbiri 157 . . 3 ({A, 𝐶} = {B, 𝐶} → B {A, 𝐶})
117elpr 3385 . . 3 (B {A, 𝐶} ↔ (B = A B = 𝐶))
1210, 11sylib 127 . 2 ({A, 𝐶} = {B, 𝐶} → (B = A B = 𝐶))
13 eqcom 2039 . 2 (A = BB = A)
14 eqeq2 2046 . 2 (A = 𝐶 → (B = AB = 𝐶))
156, 12, 13, 14oplem1 881 1 ({A, 𝐶} = {B, 𝐶} → A = B)
Colors of variables: wff set class
Syntax hints:  wi 4   wo 628   = wceq 1242   wcel 1390  Vcvv 2551  {cpr 3368
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374
This theorem is referenced by:  preqr2  3531
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