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Theorem preqr1g 3528
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 3530. (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 3465 . . . . . . 7 (A V → A {A, 𝐶})
2 eleq2 2098 . . . . . . 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 3384 . . . . . 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 3465 . . . . . . 7 (B V → B {B, 𝐶})
9 eleq2 2098 . . . . . . 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 3384 . . . . . 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 2039 . . 3 (A = BB = A)
16 eqeq2 2046 . . 3 (A = 𝐶 → (B = AB = 𝐶))
177, 14, 15, 16oplem1 881 . 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 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:  preqr2g  3529
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