Theorem en3lplem1 9571

Description: Lemma for en3lp 9573. (Contributed by Alan Sare, 28-Oct-2011.)

Assertion

Ref	Expression
en3lplem1	⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 = 𝐴 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem en3lplem1

Step	Hyp	Ref	Expression
1		simp3 1138	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐴)
2		eleq2 2818	. . 3 ⊢ (𝑥 = 𝐴 → (𝐶 ∈ 𝑥 ↔ 𝐶 ∈ 𝐴))
3	1, 2	syl5ibrcom 247	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 = 𝐴 → 𝐶 ∈ 𝑥))
4		tpid3g 4738	. . . . 5 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
5	4	3ad2ant3 1135	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
6		inelcm 4430	. . . 4 ⊢ ((𝐶 ∈ 𝑥 ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
7	5, 6	sylan2 593	. . 3 ⊢ ((𝐶 ∈ 𝑥 ∧ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
8	7	expcom 413	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝐶 ∈ 𝑥 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
9	3, 8	syld 47	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 = 𝐴 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926   ∩ cin 3915  ∅c0 4298  {ctp 4595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-nul 4299  df-sn 4592  df-pr 4594  df-tp 4596

This theorem is referenced by:  en3lplem2  9572