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Theorem kmlem4 8927
 Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.)
Assertion
Ref Expression
kmlem4 ((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅)
Distinct variable group:   𝑥,𝑤,𝑧

Proof of Theorem kmlem4
Dummy variables 𝑣 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2686 . . . . . . 7 (𝑣 = 𝑤 → (𝑣𝑥𝑤𝑥))
2 neeq2 2853 . . . . . . 7 (𝑣 = 𝑤 → (𝑧𝑣𝑧𝑤))
31, 2anbi12d 746 . . . . . 6 (𝑣 = 𝑤 → ((𝑣𝑥𝑧𝑣) ↔ (𝑤𝑥𝑧𝑤)))
4 elequ2 2001 . . . . . . 7 (𝑣 = 𝑤 → (𝑦𝑣𝑦𝑤))
54notbid 308 . . . . . 6 (𝑣 = 𝑤 → (¬ 𝑦𝑣 ↔ ¬ 𝑦𝑤))
63, 5imbi12d 334 . . . . 5 (𝑣 = 𝑤 → (((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣) ↔ ((𝑤𝑥𝑧𝑤) → ¬ 𝑦𝑤)))
76spv 2259 . . . 4 (∀𝑣((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣) → ((𝑤𝑥𝑧𝑤) → ¬ 𝑦𝑤))
8 eldif 3569 . . . . 5 (𝑦 ∈ (𝑧 (𝑥 ∖ {𝑧})) ↔ (𝑦𝑧 ∧ ¬ 𝑦 (𝑥 ∖ {𝑧})))
9 simpr 477 . . . . . 6 ((𝑦𝑧 ∧ ¬ 𝑦 (𝑥 ∖ {𝑧})) → ¬ 𝑦 (𝑥 ∖ {𝑧}))
10 eluni 4410 . . . . . . . 8 (𝑦 (𝑥 ∖ {𝑧}) ↔ ∃𝑣(𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})))
1110notbii 310 . . . . . . 7 𝑦 (𝑥 ∖ {𝑧}) ↔ ¬ ∃𝑣(𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})))
12 alnex 1703 . . . . . . 7 (∀𝑣 ¬ (𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ¬ ∃𝑣(𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})))
13 con2b 349 . . . . . . . . 9 ((𝑦𝑣 → ¬ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ (𝑣 ∈ (𝑥 ∖ {𝑧}) → ¬ 𝑦𝑣))
14 imnan 438 . . . . . . . . 9 ((𝑦𝑣 → ¬ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ¬ (𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})))
15 eldifsn 4292 . . . . . . . . . . 11 (𝑣 ∈ (𝑥 ∖ {𝑧}) ↔ (𝑣𝑥𝑣𝑧))
16 necom 2843 . . . . . . . . . . . 12 (𝑣𝑧𝑧𝑣)
1716anbi2i 729 . . . . . . . . . . 11 ((𝑣𝑥𝑣𝑧) ↔ (𝑣𝑥𝑧𝑣))
1815, 17bitri 264 . . . . . . . . . 10 (𝑣 ∈ (𝑥 ∖ {𝑧}) ↔ (𝑣𝑥𝑧𝑣))
1918imbi1i 339 . . . . . . . . 9 ((𝑣 ∈ (𝑥 ∖ {𝑧}) → ¬ 𝑦𝑣) ↔ ((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
2013, 14, 193bitr3i 290 . . . . . . . 8 (¬ (𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
2120albii 1744 . . . . . . 7 (∀𝑣 ¬ (𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ∀𝑣((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
2211, 12, 213bitr2i 288 . . . . . 6 𝑦 (𝑥 ∖ {𝑧}) ↔ ∀𝑣((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
239, 22sylib 208 . . . . 5 ((𝑦𝑧 ∧ ¬ 𝑦 (𝑥 ∖ {𝑧})) → ∀𝑣((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
248, 23sylbi 207 . . . 4 (𝑦 ∈ (𝑧 (𝑥 ∖ {𝑧})) → ∀𝑣((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
257, 24syl11 33 . . 3 ((𝑤𝑥𝑧𝑤) → (𝑦 ∈ (𝑧 (𝑥 ∖ {𝑧})) → ¬ 𝑦𝑤))
2625ralrimiv 2960 . 2 ((𝑤𝑥𝑧𝑤) → ∀𝑦 ∈ (𝑧 (𝑥 ∖ {𝑧})) ¬ 𝑦𝑤)
27 disj 3994 . 2 (((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅ ↔ ∀𝑦 ∈ (𝑧 (𝑥 ∖ {𝑧})) ¬ 𝑦𝑤)
2826, 27sylibr 224 1 ((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384  ∀wal 1478   = wceq 1480  ∃wex 1701   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907   ∖ cdif 3556   ∩ cin 3558  ∅c0 3896  {csn 4153  ∪ cuni 4407 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-v 3191  df-dif 3562  df-in 3566  df-nul 3897  df-sn 4154  df-uni 4408 This theorem is referenced by:  kmlem5  8928  kmlem11  8934
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