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Theorem elpr2 3517
 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 = 𝐵𝐴 = 𝐶))

Proof of Theorem elpr2
StepHypRef Expression
1 elprg 3515 . . 3 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
21ibi 175 . 2 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
3 elpr2.1 . . . . . 6 𝐵 ∈ V
4 eleq1 2178 . . . . . 6 (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
53, 4mpbiri 167 . . . . 5 (𝐴 = 𝐵𝐴 ∈ V)
6 elpr2.2 . . . . . 6 𝐶 ∈ V
7 eleq1 2178 . . . . . 6 (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
86, 7mpbiri 167 . . . . 5 (𝐴 = 𝐶𝐴 ∈ V)
95, 8jaoi 688 . . . 4 ((𝐴 = 𝐵𝐴 = 𝐶) → 𝐴 ∈ V)
10 elprg 3515 . . . 4 (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
119, 10syl 14 . . 3 ((𝐴 = 𝐵𝐴 = 𝐶) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
1211ibir 176 . 2 ((𝐴 = 𝐵𝐴 = 𝐶) → 𝐴 ∈ {𝐵, 𝐶})
132, 12impbii 125 1 (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   ∨ wo 680   = wceq 1314   ∈ wcel 1463  Vcvv 2658  {cpr 3496 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-sn 3501  df-pr 3502 This theorem is referenced by:  elxr  9514
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