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Theorem kmlem4 9228
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 eleq1w 2827 . . . . . . 7 (𝑣 = 𝑤 → (𝑣𝑥𝑤𝑥))
2 neeq2 3000 . . . . . . 7 (𝑣 = 𝑤 → (𝑧𝑣𝑧𝑤))
31, 2anbi12d 624 . . . . . 6 (𝑣 = 𝑤 → ((𝑣𝑥𝑧𝑣) ↔ (𝑤𝑥𝑧𝑤)))
4 elequ2 2169 . . . . . . 7 (𝑣 = 𝑤 → (𝑦𝑣𝑦𝑤))
54notbid 309 . . . . . 6 (𝑣 = 𝑤 → (¬ 𝑦𝑣 ↔ ¬ 𝑦𝑤))
63, 5imbi12d 335 . . . . 5 (𝑣 = 𝑤 → (((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣) ↔ ((𝑤𝑥𝑧𝑤) → ¬ 𝑦𝑤)))
76spv 2366 . . . 4 (∀𝑣((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣) → ((𝑤𝑥𝑧𝑤) → ¬ 𝑦𝑤))
8 eldif 3742 . . . . 5 (𝑦 ∈ (𝑧 (𝑥 ∖ {𝑧})) ↔ (𝑦𝑧 ∧ ¬ 𝑦 (𝑥 ∖ {𝑧})))
9 simpr 477 . . . . . 6 ((𝑦𝑧 ∧ ¬ 𝑦 (𝑥 ∖ {𝑧})) → ¬ 𝑦 (𝑥 ∖ {𝑧}))
10 eluni 4597 . . . . . . . 8 (𝑦 (𝑥 ∖ {𝑧}) ↔ ∃𝑣(𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})))
1110notbii 311 . . . . . . 7 𝑦 (𝑥 ∖ {𝑧}) ↔ ¬ ∃𝑣(𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})))
12 alnex 1876 . . . . . . 7 (∀𝑣 ¬ (𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ¬ ∃𝑣(𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})))
13 con2b 350 . . . . . . . . 9 ((𝑦𝑣 → ¬ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ (𝑣 ∈ (𝑥 ∖ {𝑧}) → ¬ 𝑦𝑣))
14 imnan 388 . . . . . . . . 9 ((𝑦𝑣 → ¬ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ¬ (𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})))
15 eldifsn 4472 . . . . . . . . . . 11 (𝑣 ∈ (𝑥 ∖ {𝑧}) ↔ (𝑣𝑥𝑣𝑧))
16 necom 2990 . . . . . . . . . . . 12 (𝑣𝑧𝑧𝑣)
1716anbi2i 616 . . . . . . . . . . 11 ((𝑣𝑥𝑣𝑧) ↔ (𝑣𝑥𝑧𝑣))
1815, 17bitri 266 . . . . . . . . . 10 (𝑣 ∈ (𝑥 ∖ {𝑧}) ↔ (𝑣𝑥𝑧𝑣))
1918imbi1i 340 . . . . . . . . 9 ((𝑣 ∈ (𝑥 ∖ {𝑧}) → ¬ 𝑦𝑣) ↔ ((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
2013, 14, 193bitr3i 292 . . . . . . . 8 (¬ (𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
2120albii 1914 . . . . . . 7 (∀𝑣 ¬ (𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ∀𝑣((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
2211, 12, 213bitr2i 290 . . . . . 6 𝑦 (𝑥 ∖ {𝑧}) ↔ ∀𝑣((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
239, 22sylib 209 . . . . 5 ((𝑦𝑧 ∧ ¬ 𝑦 (𝑥 ∖ {𝑧})) → ∀𝑣((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
248, 23sylbi 208 . . . 4 (𝑦 ∈ (𝑧 (𝑥 ∖ {𝑧})) → ∀𝑣((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
257, 24syl11 33 . . 3 ((𝑤𝑥𝑧𝑤) → (𝑦 ∈ (𝑧 (𝑥 ∖ {𝑧})) → ¬ 𝑦𝑤))
2625ralrimiv 3112 . 2 ((𝑤𝑥𝑧𝑤) → ∀𝑦 ∈ (𝑧 (𝑥 ∖ {𝑧})) ¬ 𝑦𝑤)
27 disj 4178 . 2 (((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅ ↔ ∀𝑦 ∈ (𝑧 (𝑥 ∖ {𝑧})) ¬ 𝑦𝑤)
2826, 27sylibr 225 1 ((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1650   = wceq 1652  wex 1874  wcel 2155  wne 2937  wral 3055  cdif 3729  cin 3731  c0 4079  {csn 4334   cuni 4594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-v 3352  df-dif 3735  df-in 3739  df-nul 4080  df-sn 4335  df-uni 4595
This theorem is referenced by:  kmlem5  9229  kmlem11  9235
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