Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elprneb Structured version   Visualization version   GIF version

Theorem elprneb 46038
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 4650 . . 3 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
2 neeq1 3003 . . . . . 6 (𝐵 = 𝐴 → (𝐵𝐶𝐴𝐶))
32eqcoms 2740 . . . . 5 (𝐴 = 𝐵 → (𝐵𝐶𝐴𝐶))
4 pm5.1 822 . . . . . 6 ((𝐴 = 𝐵𝐴𝐶) → (𝐴 = 𝐵𝐴𝐶))
54ex 413 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐶 → (𝐴 = 𝐵𝐴𝐶)))
63, 5sylbid 239 . . . 4 (𝐴 = 𝐵 → (𝐵𝐶 → (𝐴 = 𝐵𝐴𝐶)))
7 neeq2 3004 . . . . 5 (𝐴 = 𝐶 → (𝐵𝐴𝐵𝐶))
8 nesym 2997 . . . . . . . 8 (𝐵𝐴 ↔ ¬ 𝐴 = 𝐵)
9 pm5.1 822 . . . . . . . 8 ((𝐴 = 𝐶 ∧ ¬ 𝐴 = 𝐵) → (𝐴 = 𝐶 ↔ ¬ 𝐴 = 𝐵))
108, 9sylan2b 594 . . . . . . 7 ((𝐴 = 𝐶𝐵𝐴) → (𝐴 = 𝐶 ↔ ¬ 𝐴 = 𝐵))
1110necon2abid 2983 . . . . . 6 ((𝐴 = 𝐶𝐵𝐴) → (𝐴 = 𝐵𝐴𝐶))
1211ex 413 . . . . 5 (𝐴 = 𝐶 → (𝐵𝐴 → (𝐴 = 𝐵𝐴𝐶)))
137, 12sylbird 259 . . . 4 (𝐴 = 𝐶 → (𝐵𝐶 → (𝐴 = 𝐵𝐴𝐶)))
146, 13jaoi 855 . . 3 ((𝐴 = 𝐵𝐴 = 𝐶) → (𝐵𝐶 → (𝐴 = 𝐵𝐴𝐶)))
151, 14syl 17 . 2 (𝐴 ∈ {𝐵, 𝐶} → (𝐵𝐶 → (𝐴 = 𝐵𝐴𝐶)))
1615imp 407 1 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐵𝐶) → (𝐴 = 𝐵𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  wne 2940  {cpr 4630
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-v 3476  df-un 3953  df-sn 4629  df-pr 4631
This theorem is referenced by:  dfodd5  46627
  Copyright terms: Public domain W3C validator