MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  en3lplem1 Structured version   Visualization version   GIF version

Theorem en3lplem1 9063
Description: Lemma for en3lp 9065. (Contributed by Alan Sare, 28-Oct-2011.)
Assertion
Ref Expression
en3lplem1 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem en3lplem1
StepHypRef Expression
1 simp3 1135 . . 3 ((𝐴𝐵𝐵𝐶𝐶𝐴) → 𝐶𝐴)
2 eleq2 2881 . . 3 (𝑥 = 𝐴 → (𝐶𝑥𝐶𝐴))
31, 2syl5ibrcom 250 . 2 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴𝐶𝑥))
4 tpid3g 4671 . . . . 5 (𝐶𝐴𝐶 ∈ {𝐴, 𝐵, 𝐶})
543ad2ant3 1132 . . . 4 ((𝐴𝐵𝐵𝐶𝐶𝐴) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
6 inelcm 4375 . . . 4 ((𝐶𝑥𝐶 ∈ {𝐴, 𝐵, 𝐶}) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
75, 6sylan2 595 . . 3 ((𝐶𝑥 ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
87expcom 417 . 2 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐶𝑥 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
93, 8syld 47 1 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2112  wne 2990  cin 3883  c0 4246  {ctp 4532
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ne 2991  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-nul 4247  df-sn 4529  df-pr 4531  df-tp 4533
This theorem is referenced by:  en3lplem2  9064
  Copyright terms: Public domain W3C validator