Theorem en3lplem1 9567

Description: Lemma for en3lp 9569. (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 1151	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐴)
2		eleq2 2851	. . 3 ⊢ (𝑥 = 𝐴 → (𝐶 ∈ 𝑥 ↔ 𝐶 ∈ 𝐴))
3	1, 2	syl5ibrcom 249	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 = 𝐴 → 𝐶 ∈ 𝑥))
4		tpid3g 4731	. . . . 5 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
5	4	3ad2ant3 1148	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
6		inelcm 4419	. . . 4 ⊢ ((𝐶 ∈ 𝑥 ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
7	5, 6	sylan2 602	. . 3 ⊢ ((𝐶 ∈ 𝑥 ∧ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
8	7	expcom 417	. 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 1098   = wceq 1560   ∈ wcel 2142   ≠ wne 2957   ∩ cin 3903  ∅c0 4285  {ctp 4586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-nul 4286  df-sn 4583  df-pr 4585  df-tp 4587

This theorem is referenced by:  en3lplem2  9568