Theorem en3lplem1 9581

Description: Lemma for en3lp 9583. (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 1154	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐴)
2		eleq2 2858	. . 3 ⊢ (𝑥 = 𝐴 → (𝐶 ∈ 𝑥 ↔ 𝐶 ∈ 𝐴))
3	1, 2	syl5ibrcom 250	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 = 𝐴 → 𝐶 ∈ 𝑥))
4		tpid3g 4743	. . . . 5 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
5	4	3ad2ant3 1151	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
6		inelcm 4431	. . . 4 ⊢ ((𝐶 ∈ 𝑥 ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
7	5, 6	sylan2 604	. . 3 ⊢ ((𝐶 ∈ 𝑥 ∧ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
8	7	expcom 418	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝐶 ∈ 𝑥 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
9	3, 8	syld 48	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 = 𝐴 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964   ∩ cin 3912  ∅c0 4294  {ctp 4598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-nul 4295  df-sn 4595  df-pr 4597  df-tp 4599

This theorem is referenced by:  en3lplem2  9582