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Theorem elprneb 43484
 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 4571 . . 3 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
2 neeq1 3076 . . . . . 6 (𝐵 = 𝐴 → (𝐵𝐶𝐴𝐶))
32eqcoms 2832 . . . . 5 (𝐴 = 𝐵 → (𝐵𝐶𝐴𝐶))
4 pm5.1 822 . . . . . 6 ((𝐴 = 𝐵𝐴𝐶) → (𝐴 = 𝐵𝐴𝐶))
54ex 416 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐶 → (𝐴 = 𝐵𝐴𝐶)))
63, 5sylbid 243 . . . 4 (𝐴 = 𝐵 → (𝐵𝐶 → (𝐴 = 𝐵𝐴𝐶)))
7 neeq2 3077 . . . . 5 (𝐴 = 𝐶 → (𝐵𝐴𝐵𝐶))
8 nesym 3070 . . . . . . . 8 (𝐵𝐴 ↔ ¬ 𝐴 = 𝐵)
9 pm5.1 822 . . . . . . . 8 ((𝐴 = 𝐶 ∧ ¬ 𝐴 = 𝐵) → (𝐴 = 𝐶 ↔ ¬ 𝐴 = 𝐵))
108, 9sylan2b 596 . . . . . . 7 ((𝐴 = 𝐶𝐵𝐴) → (𝐴 = 𝐶 ↔ ¬ 𝐴 = 𝐵))
1110necon2abid 3056 . . . . . 6 ((𝐴 = 𝐶𝐵𝐴) → (𝐴 = 𝐵𝐴𝐶))
1211ex 416 . . . . 5 (𝐴 = 𝐶 → (𝐵𝐴 → (𝐴 = 𝐵𝐴𝐶)))
137, 12sylbird 263 . . . 4 (𝐴 = 𝐶 → (𝐵𝐶 → (𝐴 = 𝐵𝐴𝐶)))
146, 13jaoi 854 . . 3 ((𝐴 = 𝐵𝐴 = 𝐶) → (𝐵𝐶 → (𝐴 = 𝐵𝐴𝐶)))
151, 14syl 17 . 2 (𝐴 ∈ {𝐵, 𝐶} → (𝐵𝐶 → (𝐴 = 𝐵𝐴𝐶)))
1615imp 410 1 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐵𝐶) → (𝐴 = 𝐵𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2115   ≠ wne 3014  {cpr 4551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ne 3015  df-v 3482  df-un 3924  df-sn 4550  df-pr 4552 This theorem is referenced by:  dfodd5  44041
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