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Theorem nelpri 3399
 Description: If an element doesn't match the items in an unordered pair, it is not in the unordered pair. (Contributed by David A. Wheeler, 10-May-2015.)
Hypotheses
Ref Expression
nelpri.1 𝐴𝐵
nelpri.2 𝐴𝐶
Assertion
Ref Expression
nelpri ¬ 𝐴 ∈ {𝐵, 𝐶}

Proof of Theorem nelpri
StepHypRef Expression
1 nelpri.1 . 2 𝐴𝐵
2 nelpri.2 . 2 𝐴𝐶
3 neanior 2292 . . 3 ((𝐴𝐵𝐴𝐶) ↔ ¬ (𝐴 = 𝐵𝐴 = 𝐶))
4 elpri 3398 . . . 4 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
54con3i 562 . . 3 (¬ (𝐴 = 𝐵𝐴 = 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
63, 5sylbi 114 . 2 ((𝐴𝐵𝐴𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
71, 2, 6mp2an 402 1 ¬ 𝐴 ∈ {𝐵, 𝐶}
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 97   ∨ wo 629   = wceq 1243   ∈ wcel 1393   ≠ wne 2204  {cpr 3376 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-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382 This theorem is referenced by: (None)
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