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Theorem elpr2 3369
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 3367 . . 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 2082 . . . . . 6 (A = B → (A V ↔ B V))
53, 4mpbiri 157 . . . . 5 (A = BA V)
6 elpr2.2 . . . . . 6 𝐶 V
7 eleq1 2082 . . . . . 6 (A = 𝐶 → (A V ↔ 𝐶 V))
86, 7mpbiri 157 . . . . 5 (A = 𝐶A V)
95, 8jaoi 623 . . . 4 ((A = B A = 𝐶) → A V)
10 elprg 3367 . . . 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 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: (None)
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