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Theorem en3lplem1 8722
Description: Lemma for en3lp 8724. (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 1168 . . 3 ((𝐴𝐵𝐵𝐶𝐶𝐴) → 𝐶𝐴)
2 eleq2 2833 . . 3 (𝑥 = 𝐴 → (𝐶𝑥𝐶𝐴))
31, 2syl5ibrcom 238 . 2 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴𝐶𝑥))
4 tpid3g 4460 . . . . 5 (𝐶𝐴𝐶 ∈ {𝐴, 𝐵, 𝐶})
543ad2ant3 1165 . . . 4 ((𝐴𝐵𝐵𝐶𝐶𝐴) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
6 inelcm 4193 . . . 4 ((𝐶𝑥𝐶 ∈ {𝐴, 𝐵, 𝐶}) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
75, 6sylan2 586 . . 3 ((𝐶𝑥 ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
87expcom 402 . 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 1107   = wceq 1652  wcel 2155  wne 2937  cin 3731  c0 4079  {ctp 4338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-nul 4080  df-sn 4335  df-pr 4337  df-tp 4339
This theorem is referenced by:  en3lplem2  8723
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