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Theorem elprneb 39939
 Description: An element of a proper unordered pair is the first element iff it is not the second element. (Contributed by AV, 18-Jun-2020.)
Assertion
Ref Expression
elprneb ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐵𝐶) → (𝐴 = 𝐵𝐴𝐶))

Proof of Theorem elprneb
StepHypRef Expression
1 elpri 4145 . . 3 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
2 neeq1 2844 . . . . . 6 (𝐵 = 𝐴 → (𝐵𝐶𝐴𝐶))
32eqcoms 2618 . . . . 5 (𝐴 = 𝐵 → (𝐵𝐶𝐴𝐶))
4 pm5.1 898 . . . . . 6 ((𝐴 = 𝐵𝐴𝐶) → (𝐴 = 𝐵𝐴𝐶))
54ex 449 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐶 → (𝐴 = 𝐵𝐴𝐶)))
63, 5sylbid 229 . . . 4 (𝐴 = 𝐵 → (𝐵𝐶 → (𝐴 = 𝐵𝐴𝐶)))
7 neeq2 2845 . . . . 5 (𝐴 = 𝐶 → (𝐵𝐴𝐵𝐶))
8 nesym 2838 . . . . . . . 8 (𝐵𝐴 ↔ ¬ 𝐴 = 𝐵)
9 pm5.1 898 . . . . . . . 8 ((𝐴 = 𝐶 ∧ ¬ 𝐴 = 𝐵) → (𝐴 = 𝐶 ↔ ¬ 𝐴 = 𝐵))
108, 9sylan2b 491 . . . . . . 7 ((𝐴 = 𝐶𝐵𝐴) → (𝐴 = 𝐶 ↔ ¬ 𝐴 = 𝐵))
1110necon2abid 2824 . . . . . 6 ((𝐴 = 𝐶𝐵𝐴) → (𝐴 = 𝐵𝐴𝐶))
1211ex 449 . . . . 5 (𝐴 = 𝐶 → (𝐵𝐴 → (𝐴 = 𝐵𝐴𝐶)))
137, 12sylbird 249 . . . 4 (𝐴 = 𝐶 → (𝐵𝐶 → (𝐴 = 𝐵𝐴𝐶)))
146, 13jaoi 393 . . 3 ((𝐴 = 𝐵𝐴 = 𝐶) → (𝐵𝐶 → (𝐴 = 𝐵𝐴𝐶)))
151, 14syl 17 . 2 (𝐴 ∈ {𝐵, 𝐶} → (𝐵𝐶 → (𝐴 = 𝐵𝐴𝐶)))
1615imp 444 1 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐵𝐶) → (𝐴 = 𝐵𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {cpr 4127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128 This theorem is referenced by:  dfodd5  40110
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