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Theorem elpr2 3386
Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.)
Hypotheses
Ref Expression
elpr2.1 B V
elpr2.2 𝐶 V
Assertion
Ref Expression
elpr2 (A {B, 𝐶} ↔ (A = B A = 𝐶))

Proof of Theorem elpr2
StepHypRef Expression
1 elprg 3384 . . 3 (A {B, 𝐶} → (A {B, 𝐶} ↔ (A = B A = 𝐶)))
21ibi 165 . 2 (A {B, 𝐶} → (A = B A = 𝐶))
3 elpr2.1 . . . . . 6 B V
4 eleq1 2097 . . . . . 6 (A = B → (A V ↔ B V))
53, 4mpbiri 157 . . . . 5 (A = BA V)
6 elpr2.2 . . . . . 6 𝐶 V
7 eleq1 2097 . . . . . 6 (A = 𝐶 → (A V ↔ 𝐶 V))
86, 7mpbiri 157 . . . . 5 (A = 𝐶A V)
95, 8jaoi 635 . . . 4 ((A = B A = 𝐶) → A V)
10 elprg 3384 . . . 4 (A V → (A {B, 𝐶} ↔ (A = B A = 𝐶)))
119, 10syl 14 . . 3 ((A = B A = 𝐶) → (A {B, 𝐶} ↔ (A = B A = 𝐶)))
1211ibir 166 . 2 ((A = B A = 𝐶) → A {B, 𝐶})
132, 12impbii 117 1 (A {B, 𝐶} ↔ (A = B A = 𝐶))
Colors of variables: wff set class
Syntax hints:  wb 98   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:  elxr  8426
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