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Theorem elprg 3422
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 2062 . . 3 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
2 eqeq1 2062 . . 3 (𝑥 = 𝐴 → (𝑥 = 𝐶𝐴 = 𝐶))
31, 2orbi12d 717 . 2 (𝑥 = 𝐴 → ((𝑥 = 𝐵𝑥 = 𝐶) ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
4 dfpr2 3421 . 2 {𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐵𝑥 = 𝐶)}
53, 4elab2g 2711 1 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wo 639   = wceq 1259  wcel 1409  {cpr 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-sn 3408  df-pr 3409
This theorem is referenced by:  elpr  3423  elpr2  3424  elpri  3425  eltpg  3443  prid1g  3501  preqr1g  3564  m1expeven  9466
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