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Theorem kmlem4 9567
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 elequ1 2112 . . . . . . 7 (𝑣 = 𝑤 → (𝑣𝑥𝑤𝑥))
2 neeq2 3076 . . . . . . 7 (𝑣 = 𝑤 → (𝑧𝑣𝑧𝑤))
31, 2anbi12d 630 . . . . . 6 (𝑣 = 𝑤 → ((𝑣𝑥𝑧𝑣) ↔ (𝑤𝑥𝑧𝑤)))
4 elequ2 2120 . . . . . . 7 (𝑣 = 𝑤 → (𝑦𝑣𝑦𝑤))
54notbid 319 . . . . . 6 (𝑣 = 𝑤 → (¬ 𝑦𝑣 ↔ ¬ 𝑦𝑤))
63, 5imbi12d 346 . . . . 5 (𝑣 = 𝑤 → (((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣) ↔ ((𝑤𝑥𝑧𝑤) → ¬ 𝑦𝑤)))
76spvv 1994 . . . 4 (∀𝑣((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣) → ((𝑤𝑥𝑧𝑤) → ¬ 𝑦𝑤))
8 eldif 3943 . . . . 5 (𝑦 ∈ (𝑧 (𝑥 ∖ {𝑧})) ↔ (𝑦𝑧 ∧ ¬ 𝑦 (𝑥 ∖ {𝑧})))
9 simpr 485 . . . . . 6 ((𝑦𝑧 ∧ ¬ 𝑦 (𝑥 ∖ {𝑧})) → ¬ 𝑦 (𝑥 ∖ {𝑧}))
10 eluni 4833 . . . . . . . 8 (𝑦 (𝑥 ∖ {𝑧}) ↔ ∃𝑣(𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})))
1110notbii 321 . . . . . . 7 𝑦 (𝑥 ∖ {𝑧}) ↔ ¬ ∃𝑣(𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})))
12 alnex 1773 . . . . . . 7 (∀𝑣 ¬ (𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ¬ ∃𝑣(𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})))
13 con2b 361 . . . . . . . . 9 ((𝑦𝑣 → ¬ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ (𝑣 ∈ (𝑥 ∖ {𝑧}) → ¬ 𝑦𝑣))
14 imnan 400 . . . . . . . . 9 ((𝑦𝑣 → ¬ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ¬ (𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})))
15 eldifsn 4711 . . . . . . . . . . 11 (𝑣 ∈ (𝑥 ∖ {𝑧}) ↔ (𝑣𝑥𝑣𝑧))
16 necom 3066 . . . . . . . . . . . 12 (𝑣𝑧𝑧𝑣)
1716anbi2i 622 . . . . . . . . . . 11 ((𝑣𝑥𝑣𝑧) ↔ (𝑣𝑥𝑧𝑣))
1815, 17bitri 276 . . . . . . . . . 10 (𝑣 ∈ (𝑥 ∖ {𝑧}) ↔ (𝑣𝑥𝑧𝑣))
1918imbi1i 351 . . . . . . . . 9 ((𝑣 ∈ (𝑥 ∖ {𝑧}) → ¬ 𝑦𝑣) ↔ ((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
2013, 14, 193bitr3i 302 . . . . . . . 8 (¬ (𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
2120albii 1811 . . . . . . 7 (∀𝑣 ¬ (𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ∀𝑣((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
2211, 12, 213bitr2i 300 . . . . . 6 𝑦 (𝑥 ∖ {𝑧}) ↔ ∀𝑣((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
239, 22sylib 219 . . . . 5 ((𝑦𝑧 ∧ ¬ 𝑦 (𝑥 ∖ {𝑧})) → ∀𝑣((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
248, 23sylbi 218 . . . 4 (𝑦 ∈ (𝑧 (𝑥 ∖ {𝑧})) → ∀𝑣((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
257, 24syl11 33 . . 3 ((𝑤𝑥𝑧𝑤) → (𝑦 ∈ (𝑧 (𝑥 ∖ {𝑧})) → ¬ 𝑦𝑤))
2625ralrimiv 3178 . 2 ((𝑤𝑥𝑧𝑤) → ∀𝑦 ∈ (𝑧 (𝑥 ∖ {𝑧})) ¬ 𝑦𝑤)
27 disj 4395 . 2 (((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅ ↔ ∀𝑦 ∈ (𝑧 (𝑥 ∖ {𝑧})) ¬ 𝑦𝑤)
2826, 27sylibr 235 1 ((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wal 1526   = wceq 1528  wex 1771  wcel 2105  wne 3013  wral 3135  cdif 3930  cin 3932  c0 4288  {csn 4557   cuni 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-v 3494  df-dif 3936  df-in 3940  df-nul 4289  df-sn 4558  df-uni 4831
This theorem is referenced by:  kmlem5  9568  kmlem11  9574
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