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Theorem elprg 3542
 Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM, 13-Sep-1995.)
Assertion
Ref Expression
elprg (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))

Proof of Theorem elprg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2144 . . 3 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
2 eqeq1 2144 . . 3 (𝑥 = 𝐴 → (𝑥 = 𝐶𝐴 = 𝐶))
31, 2orbi12d 782 . 2 (𝑥 = 𝐴 → ((𝑥 = 𝐵𝑥 = 𝐶) ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
4 dfpr2 3541 . 2 {𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐵𝑥 = 𝐶)}
53, 4elab2g 2826 1 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   ∨ wo 697   = wceq 1331   ∈ wcel 1480  {cpr 3523 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529 This theorem is referenced by:  elpr  3543  elpr2  3544  elpri  3545  eltpg  3564  prid1g  3622  preqr1g  3688  m1expeven  10333  maxclpr  10987  minmax  10994  minclpr  11001  xrminmax  11027
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