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Theorem elprg 4585
 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 2830 . . 3 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
2 eqeq1 2830 . . 3 (𝑥 = 𝐴 → (𝑥 = 𝐶𝐴 = 𝐶))
31, 2orbi12d 914 . 2 (𝑥 = 𝐴 → ((𝑥 = 𝐵𝑥 = 𝐶) ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
4 dfpr2 4583 . 2 {𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐵𝑥 = 𝐶)}
53, 4elab2g 3673 1 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∨ wo 843   = wceq 1530   ∈ wcel 2107  {cpr 4566 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-v 3502  df-un 3945  df-sn 4565  df-pr 4567 This theorem is referenced by:  elpri  4586  elpr  4587  elpr2  4588  nelpr2  4589  nelpr1  4590  eldifpr  4594  eltpg  4622  ifpr  4628  prid1g  4695  ssprss  4756  preq1b  4776  prel12g  4793  ordunpr  7529  hashtpg  13833  2nsgsimpgd  19144  cnsubrg  20521  atandm  25367  1egrvtxdg0  27207  eupth2lem1  27911  nelpr  30205  eliccioo  30521  linds2eq  30855  sfprmdvdsmersenne  43600  prelrrx2b  44533
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