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Theorem elprneb 41065
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 4195 . . 3 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
2 neeq1 2855 . . . . . 6 (𝐵 = 𝐴 → (𝐵𝐶𝐴𝐶))
32eqcoms 2629 . . . . 5 (𝐴 = 𝐵 → (𝐵𝐶𝐴𝐶))
4 pm5.1 902 . . . . . 6 ((𝐴 = 𝐵𝐴𝐶) → (𝐴 = 𝐵𝐴𝐶))
54ex 450 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐶 → (𝐴 = 𝐵𝐴𝐶)))
63, 5sylbid 230 . . . 4 (𝐴 = 𝐵 → (𝐵𝐶 → (𝐴 = 𝐵𝐴𝐶)))
7 neeq2 2856 . . . . 5 (𝐴 = 𝐶 → (𝐵𝐴𝐵𝐶))
8 nesym 2849 . . . . . . . 8 (𝐵𝐴 ↔ ¬ 𝐴 = 𝐵)
9 pm5.1 902 . . . . . . . 8 ((𝐴 = 𝐶 ∧ ¬ 𝐴 = 𝐵) → (𝐴 = 𝐶 ↔ ¬ 𝐴 = 𝐵))
108, 9sylan2b 492 . . . . . . 7 ((𝐴 = 𝐶𝐵𝐴) → (𝐴 = 𝐶 ↔ ¬ 𝐴 = 𝐵))
1110necon2abid 2835 . . . . . 6 ((𝐴 = 𝐶𝐵𝐴) → (𝐴 = 𝐵𝐴𝐶))
1211ex 450 . . . . 5 (𝐴 = 𝐶 → (𝐵𝐴 → (𝐴 = 𝐵𝐴𝐶)))
137, 12sylbird 250 . . . 4 (𝐴 = 𝐶 → (𝐵𝐶 → (𝐴 = 𝐵𝐴𝐶)))
146, 13jaoi 394 . . 3 ((𝐴 = 𝐵𝐴 = 𝐶) → (𝐵𝐶 → (𝐴 = 𝐵𝐴𝐶)))
151, 14syl 17 . 2 (𝐴 ∈ {𝐵, 𝐶} → (𝐵𝐶 → (𝐴 = 𝐵𝐴𝐶)))
1615imp 445 1 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐵𝐶) → (𝐴 = 𝐵𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1482  wcel 1989  wne 2793  {cpr 4177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-v 3200  df-un 3577  df-sn 4176  df-pr 4178
This theorem is referenced by:  dfodd5  41343
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