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Theorem en3lplem1 8680
Description: Lemma for en3lp 8682. (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 1133 . . 3 ((𝐴𝐵𝐵𝐶𝐶𝐴) → 𝐶𝐴)
2 eleq2 2828 . . 3 (𝑥 = 𝐴 → (𝐶𝑥𝐶𝐴))
31, 2syl5ibrcom 237 . 2 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴𝐶𝑥))
4 tpid3g 4449 . . . . 5 (𝐶𝐴𝐶 ∈ {𝐴, 𝐵, 𝐶})
543ad2ant3 1130 . . . 4 ((𝐴𝐵𝐵𝐶𝐶𝐴) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
6 inelcm 4176 . . . 4 ((𝐶𝑥𝐶 ∈ {𝐴, 𝐵, 𝐶}) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
75, 6sylan2 492 . . 3 ((𝐶𝑥 ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
87expcom 450 . 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 1072   = wceq 1632  wcel 2139  wne 2932  cin 3714  c0 4058  {ctp 4325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-nul 4059  df-sn 4322  df-pr 4324  df-tp 4326
This theorem is referenced by:  en3lplem2  8681
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