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Theorem en3lplem2 8681
Description: Lemma for en3lp 8682. (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 8680 . . . . 5 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
2 en3lplem1 8680 . . . . . . . 8 ((𝐵𝐶𝐶𝐴𝐴𝐵) → (𝑥 = 𝐵 → (𝑥 ∩ {𝐵, 𝐶, 𝐴}) ≠ ∅))
323comr 1120 . . . . . . 7 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐵 → (𝑥 ∩ {𝐵, 𝐶, 𝐴}) ≠ ∅))
43a1d 25 . . . . . 6 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 = 𝐵 → (𝑥 ∩ {𝐵, 𝐶, 𝐴}) ≠ ∅)))
5 tprot 4428 . . . . . . . . 9 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
65ineq2i 3954 . . . . . . . 8 (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = (𝑥 ∩ {𝐵, 𝐶, 𝐴})
76neeq1i 2996 . . . . . . 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 535 . . . . 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 8680 . . . . . . 7 ((𝐶𝐴𝐴𝐵𝐵𝐶) → (𝑥 = 𝐶 → (𝑥 ∩ {𝐶, 𝐴, 𝐵}) ≠ ∅))
14133coml 1122 . . . . . 6 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐶 → (𝑥 ∩ {𝐶, 𝐴, 𝐵}) ≠ ∅))
1514a1d 25 . . . . 5 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 = 𝐶 → (𝑥 ∩ {𝐶, 𝐴, 𝐵}) ≠ ∅)))
16 tprot 4428 . . . . . . 7 {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
1716ineq2i 3954 . . . . . 6 (𝑥 ∩ {𝐶, 𝐴, 𝐵}) = (𝑥 ∩ {𝐴, 𝐵, 𝐶})
1817neeq1i 2996 . . . . 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 4375 . . . . . . . 8 {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)}
2322eleq2i 2831 . . . . . . 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 2748 . . . . . 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 1073 . . . . 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 1071  w3a 1072   = wceq 1632  wcel 2139  {cab 2746  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:  en3lp  8682
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