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Theorem en3lplem2 8550
 Description: Lemma for en3lp 8551. (Contributed by Alan Sare, 28-Oct-2011.)
Assertion
Ref Expression
en3lplem2 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem en3lplem2
StepHypRef Expression
1 en3lplem1 8549 . . . . 5 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
2 en3lplem1 8549 . . . . . . . 8 ((𝐵𝐶𝐶𝐴𝐴𝐵) → (𝑥 = 𝐵 → (𝑥 ∩ {𝐵, 𝐶, 𝐴}) ≠ ∅))
323comr 1290 . . . . . . 7 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐵 → (𝑥 ∩ {𝐵, 𝐶, 𝐴}) ≠ ∅))
43a1d 25 . . . . . 6 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 = 𝐵 → (𝑥 ∩ {𝐵, 𝐶, 𝐴}) ≠ ∅)))
5 tprot 4316 . . . . . . . . 9 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
65ineq2i 3844 . . . . . . . 8 (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = (𝑥 ∩ {𝐵, 𝐶, 𝐴})
76neeq1i 2887 . . . . . . 7 ((𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅ ↔ (𝑥 ∩ {𝐵, 𝐶, 𝐴}) ≠ ∅)
87bicomi 214 . . . . . 6 ((𝑥 ∩ {𝐵, 𝐶, 𝐴}) ≠ ∅ ↔ (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
94, 8syl8ib 246 . . . . 5 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 = 𝐵 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)))
10 jao 533 . . . . 5 ((𝑥 = 𝐴 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅) → ((𝑥 = 𝐵 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅) → ((𝑥 = 𝐴𝑥 = 𝐵) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)))
111, 9, 10sylsyld 61 . . . 4 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ((𝑥 = 𝐴𝑥 = 𝐵) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)))
1211imp 444 . . 3 (((𝐴𝐵𝐵𝐶𝐶𝐴) ∧ 𝑥 ∈ {𝐴, 𝐵, 𝐶}) → ((𝑥 = 𝐴𝑥 = 𝐵) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
13 en3lplem1 8549 . . . . . . 7 ((𝐶𝐴𝐴𝐵𝐵𝐶) → (𝑥 = 𝐶 → (𝑥 ∩ {𝐶, 𝐴, 𝐵}) ≠ ∅))
14133coml 1292 . . . . . 6 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐶 → (𝑥 ∩ {𝐶, 𝐴, 𝐵}) ≠ ∅))
1514a1d 25 . . . . 5 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 = 𝐶 → (𝑥 ∩ {𝐶, 𝐴, 𝐵}) ≠ ∅)))
16 tprot 4316 . . . . . . 7 {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
1716ineq2i 3844 . . . . . 6 (𝑥 ∩ {𝐶, 𝐴, 𝐵}) = (𝑥 ∩ {𝐴, 𝐵, 𝐶})
1817neeq1i 2887 . . . . 5 ((𝑥 ∩ {𝐶, 𝐴, 𝐵}) ≠ ∅ ↔ (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
1915, 18syl8ib 246 . . . 4 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 = 𝐶 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)))
2019imp 444 . . 3 (((𝐴𝐵𝐵𝐶𝐶𝐴) ∧ 𝑥 ∈ {𝐴, 𝐵, 𝐶}) → (𝑥 = 𝐶 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
21 idd 24 . . . . . . 7 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → 𝑥 ∈ {𝐴, 𝐵, 𝐶}))
22 dftp2 4263 . . . . . . . 8 {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)}
2322eleq2i 2722 . . . . . . 7 (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)})
2421, 23syl6ib 241 . . . . . 6 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)}))
25 abid 2639 . . . . . 6 (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)} ↔ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶))
2624, 25syl6ib 241 . . . . 5 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)))
27 df-3or 1055 . . . . 5 ((𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶) ↔ ((𝑥 = 𝐴𝑥 = 𝐵) ∨ 𝑥 = 𝐶))
2826, 27syl6ib 241 . . . 4 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ((𝑥 = 𝐴𝑥 = 𝐵) ∨ 𝑥 = 𝐶)))
2928imp 444 . . 3 (((𝐴𝐵𝐵𝐶𝐶𝐴) ∧ 𝑥 ∈ {𝐴, 𝐵, 𝐶}) → ((𝑥 = 𝐴𝑥 = 𝐵) ∨ 𝑥 = 𝐶))
3012, 20, 29mpjaod 395 . 2 (((𝐴𝐵𝐵𝐶𝐶𝐴) ∧ 𝑥 ∈ {𝐴, 𝐵, 𝐶}) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
3130ex 449 1 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   ∨ w3o 1053   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  {cab 2637   ≠ wne 2823   ∩ cin 3606  ∅c0 3948  {ctp 4214 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-nul 3949  df-sn 4211  df-pr 4213  df-tp 4215 This theorem is referenced by:  en3lp  8551
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