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Theorem en3lplem2 9653
Description: Lemma for en3lp 9654. (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 9652 . . . . 5 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
2 en3lplem1 9652 . . . . . . . 8 ((𝐵𝐶𝐶𝐴𝐴𝐵) → (𝑥 = 𝐵 → (𝑥 ∩ {𝐵, 𝐶, 𝐴}) ≠ ∅))
323comr 1126 . . . . . . 7 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐵 → (𝑥 ∩ {𝐵, 𝐶, 𝐴}) ≠ ∅))
43a1d 25 . . . . . 6 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 = 𝐵 → (𝑥 ∩ {𝐵, 𝐶, 𝐴}) ≠ ∅)))
5 tprot 4749 . . . . . . . . 9 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
65ineq2i 4217 . . . . . . . 8 (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = (𝑥 ∩ {𝐵, 𝐶, 𝐴})
76neeq1i 3005 . . . . . . 7 ((𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅ ↔ (𝑥 ∩ {𝐵, 𝐶, 𝐴}) ≠ ∅)
87bicomi 224 . . . . . 6 ((𝑥 ∩ {𝐵, 𝐶, 𝐴}) ≠ ∅ ↔ (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
94, 8syl8ib 256 . . . . 5 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 = 𝐵 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)))
10 jao 963 . . . . 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 406 . . 3 (((𝐴𝐵𝐵𝐶𝐶𝐴) ∧ 𝑥 ∈ {𝐴, 𝐵, 𝐶}) → ((𝑥 = 𝐴𝑥 = 𝐵) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
13 en3lplem1 9652 . . . . . . 7 ((𝐶𝐴𝐴𝐵𝐵𝐶) → (𝑥 = 𝐶 → (𝑥 ∩ {𝐶, 𝐴, 𝐵}) ≠ ∅))
14133coml 1128 . . . . . 6 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐶 → (𝑥 ∩ {𝐶, 𝐴, 𝐵}) ≠ ∅))
1514a1d 25 . . . . 5 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 = 𝐶 → (𝑥 ∩ {𝐶, 𝐴, 𝐵}) ≠ ∅)))
16 tprot 4749 . . . . . . 7 {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
1716ineq2i 4217 . . . . . 6 (𝑥 ∩ {𝐶, 𝐴, 𝐵}) = (𝑥 ∩ {𝐴, 𝐵, 𝐶})
1817neeq1i 3005 . . . . 5 ((𝑥 ∩ {𝐶, 𝐴, 𝐵}) ≠ ∅ ↔ (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
1915, 18syl8ib 256 . . . 4 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 = 𝐶 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)))
2019imp 406 . . 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 4691 . . . . . . . 8 {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)}
2322eleq2i 2833 . . . . . . 7 (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)})
2421, 23imbitrdi 251 . . . . . 6 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)}))
25 abid 2718 . . . . . 6 (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)} ↔ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶))
2624, 25imbitrdi 251 . . . . 5 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)))
27 df-3or 1088 . . . . 5 ((𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶) ↔ ((𝑥 = 𝐴𝑥 = 𝐵) ∨ 𝑥 = 𝐶))
2826, 27imbitrdi 251 . . . 4 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ((𝑥 = 𝐴𝑥 = 𝐵) ∨ 𝑥 = 𝐶)))
2928imp 406 . . 3 (((𝐴𝐵𝐵𝐶𝐶𝐴) ∧ 𝑥 ∈ {𝐴, 𝐵, 𝐶}) → ((𝑥 = 𝐴𝑥 = 𝐵) ∨ 𝑥 = 𝐶))
3012, 20, 29mpjaod 861 . 2 (((𝐴𝐵𝐵𝐶𝐶𝐴) ∧ 𝑥 ∈ {𝐴, 𝐵, 𝐶}) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
3130ex 412 1 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848  w3o 1086  w3a 1087   = wceq 1540  wcel 2108  {cab 2714  wne 2940  cin 3950  c0 4333  {ctp 4630
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 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-nul 4334  df-sn 4627  df-pr 4629  df-tp 4631
This theorem is referenced by:  en3lp  9654
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