MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  kmlem4 Structured version   Visualization version   GIF version

Theorem kmlem4 10096
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 2114 . . . . . . 7 (𝑣 = 𝑤 → (𝑣𝑥𝑤𝑥))
2 neeq2 3008 . . . . . . 7 (𝑣 = 𝑤 → (𝑧𝑣𝑧𝑤))
31, 2anbi12d 632 . . . . . 6 (𝑣 = 𝑤 → ((𝑣𝑥𝑧𝑣) ↔ (𝑤𝑥𝑧𝑤)))
4 elequ2 2122 . . . . . . 7 (𝑣 = 𝑤 → (𝑦𝑣𝑦𝑤))
54notbid 318 . . . . . 6 (𝑣 = 𝑤 → (¬ 𝑦𝑣 ↔ ¬ 𝑦𝑤))
63, 5imbi12d 345 . . . . 5 (𝑣 = 𝑤 → (((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣) ↔ ((𝑤𝑥𝑧𝑤) → ¬ 𝑦𝑤)))
76spvv 2001 . . . 4 (∀𝑣((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣) → ((𝑤𝑥𝑧𝑤) → ¬ 𝑦𝑤))
8 eldif 3925 . . . . 5 (𝑦 ∈ (𝑧 (𝑥 ∖ {𝑧})) ↔ (𝑦𝑧 ∧ ¬ 𝑦 (𝑥 ∖ {𝑧})))
9 simpr 486 . . . . . 6 ((𝑦𝑧 ∧ ¬ 𝑦 (𝑥 ∖ {𝑧})) → ¬ 𝑦 (𝑥 ∖ {𝑧}))
10 eluni 4873 . . . . . . . 8 (𝑦 (𝑥 ∖ {𝑧}) ↔ ∃𝑣(𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})))
1110notbii 320 . . . . . . 7 𝑦 (𝑥 ∖ {𝑧}) ↔ ¬ ∃𝑣(𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})))
12 alnex 1784 . . . . . . 7 (∀𝑣 ¬ (𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ¬ ∃𝑣(𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})))
13 con2b 360 . . . . . . . . 9 ((𝑦𝑣 → ¬ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ (𝑣 ∈ (𝑥 ∖ {𝑧}) → ¬ 𝑦𝑣))
14 imnan 401 . . . . . . . . 9 ((𝑦𝑣 → ¬ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ¬ (𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})))
15 eldifsn 4752 . . . . . . . . . . 11 (𝑣 ∈ (𝑥 ∖ {𝑧}) ↔ (𝑣𝑥𝑣𝑧))
16 necom 2998 . . . . . . . . . . . 12 (𝑣𝑧𝑧𝑣)
1716anbi2i 624 . . . . . . . . . . 11 ((𝑣𝑥𝑣𝑧) ↔ (𝑣𝑥𝑧𝑣))
1815, 17bitri 275 . . . . . . . . . 10 (𝑣 ∈ (𝑥 ∖ {𝑧}) ↔ (𝑣𝑥𝑧𝑣))
1918imbi1i 350 . . . . . . . . 9 ((𝑣 ∈ (𝑥 ∖ {𝑧}) → ¬ 𝑦𝑣) ↔ ((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
2013, 14, 193bitr3i 301 . . . . . . . 8 (¬ (𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
2120albii 1822 . . . . . . 7 (∀𝑣 ¬ (𝑦𝑣𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ∀𝑣((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
2211, 12, 213bitr2i 299 . . . . . 6 𝑦 (𝑥 ∖ {𝑧}) ↔ ∀𝑣((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
239, 22sylib 217 . . . . 5 ((𝑦𝑧 ∧ ¬ 𝑦 (𝑥 ∖ {𝑧})) → ∀𝑣((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
248, 23sylbi 216 . . . 4 (𝑦 ∈ (𝑧 (𝑥 ∖ {𝑧})) → ∀𝑣((𝑣𝑥𝑧𝑣) → ¬ 𝑦𝑣))
257, 24syl11 33 . . 3 ((𝑤𝑥𝑧𝑤) → (𝑦 ∈ (𝑧 (𝑥 ∖ {𝑧})) → ¬ 𝑦𝑤))
2625ralrimiv 3143 . 2 ((𝑤𝑥𝑧𝑤) → ∀𝑦 ∈ (𝑧 (𝑥 ∖ {𝑧})) ¬ 𝑦𝑤)
27 disj 4412 . 2 (((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅ ↔ ∀𝑦 ∈ (𝑧 (𝑥 ∖ {𝑧})) ¬ 𝑦𝑤)
2826, 27sylibr 233 1 ((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wal 1540   = wceq 1542  wex 1782  wcel 2107  wne 2944  wral 3065  cdif 3912  cin 3914  c0 4287  {csn 4591   cuni 4870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-v 3450  df-dif 3918  df-in 3922  df-nul 4288  df-sn 4592  df-uni 4871
This theorem is referenced by:  kmlem5  10097  kmlem11  10103
  Copyright terms: Public domain W3C validator