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Theorem kmlem3 8926
Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. The right-hand side is part of the hypothesis of 4. (Contributed by NM, 25-Mar-2004.)
Assertion
Ref Expression
kmlem3 ((𝑧 (𝑥 ∖ {𝑧})) ≠ ∅ ↔ ∃𝑣𝑧𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)))
Distinct variable group:   𝑥,𝑣,𝑤,𝑧

Proof of Theorem kmlem3
StepHypRef Expression
1 dfdif2 3568 . . . 4 (𝑧 (𝑥 ∖ {𝑧})) = {𝑣𝑧 ∣ ¬ 𝑣 (𝑥 ∖ {𝑧})}
2 dfnul3 3899 . . . . . 6 ∅ = {𝑣𝑧 ∣ ¬ 𝑣𝑧}
32uneq2i 3747 . . . . 5 ({𝑣𝑧 ∣ ¬ 𝑣 (𝑥 ∖ {𝑧})} ∪ ∅) = ({𝑣𝑧 ∣ ¬ 𝑣 (𝑥 ∖ {𝑧})} ∪ {𝑣𝑧 ∣ ¬ 𝑣𝑧})
4 un0 3944 . . . . 5 ({𝑣𝑧 ∣ ¬ 𝑣 (𝑥 ∖ {𝑧})} ∪ ∅) = {𝑣𝑧 ∣ ¬ 𝑣 (𝑥 ∖ {𝑧})}
5 unrab 3879 . . . . 5 ({𝑣𝑧 ∣ ¬ 𝑣 (𝑥 ∖ {𝑧})} ∪ {𝑣𝑧 ∣ ¬ 𝑣𝑧}) = {𝑣𝑧 ∣ (¬ 𝑣 (𝑥 ∖ {𝑧}) ∨ ¬ 𝑣𝑧)}
63, 4, 53eqtr3i 2651 . . . 4 {𝑣𝑧 ∣ ¬ 𝑣 (𝑥 ∖ {𝑧})} = {𝑣𝑧 ∣ (¬ 𝑣 (𝑥 ∖ {𝑧}) ∨ ¬ 𝑣𝑧)}
7 ianor 509 . . . . . . 7 (¬ (𝑣 (𝑥 ∖ {𝑧}) ∧ 𝑣𝑧) ↔ (¬ 𝑣 (𝑥 ∖ {𝑧}) ∨ ¬ 𝑣𝑧))
8 eluni 4410 . . . . . . . . . 10 (𝑣 (𝑥 ∖ {𝑧}) ↔ ∃𝑤(𝑣𝑤𝑤 ∈ (𝑥 ∖ {𝑧})))
98anbi1i 730 . . . . . . . . 9 ((𝑣 (𝑥 ∖ {𝑧}) ∧ 𝑣𝑧) ↔ (∃𝑤(𝑣𝑤𝑤 ∈ (𝑥 ∖ {𝑧})) ∧ 𝑣𝑧))
10 df-rex 2913 . . . . . . . . . 10 (∃𝑤𝑥 ¬ (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)) ↔ ∃𝑤(𝑤𝑥 ∧ ¬ (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤))))
11 elin 3779 . . . . . . . . . . . . . . 15 (𝑣 ∈ (𝑧𝑤) ↔ (𝑣𝑧𝑣𝑤))
1211anbi2i 729 . . . . . . . . . . . . . 14 ((𝑧𝑤𝑣 ∈ (𝑧𝑤)) ↔ (𝑧𝑤 ∧ (𝑣𝑧𝑣𝑤)))
13 df-an 386 . . . . . . . . . . . . . 14 ((𝑧𝑤𝑣 ∈ (𝑧𝑤)) ↔ ¬ (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)))
1412, 13bitr3i 266 . . . . . . . . . . . . 13 ((𝑧𝑤 ∧ (𝑣𝑧𝑣𝑤)) ↔ ¬ (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)))
1514anbi2i 729 . . . . . . . . . . . 12 ((𝑤𝑥 ∧ (𝑧𝑤 ∧ (𝑣𝑧𝑣𝑤))) ↔ (𝑤𝑥 ∧ ¬ (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤))))
16 eldifsn 4292 . . . . . . . . . . . . . . . 16 (𝑤 ∈ (𝑥 ∖ {𝑧}) ↔ (𝑤𝑥𝑤𝑧))
17 necom 2843 . . . . . . . . . . . . . . . . 17 (𝑤𝑧𝑧𝑤)
1817anbi2i 729 . . . . . . . . . . . . . . . 16 ((𝑤𝑥𝑤𝑧) ↔ (𝑤𝑥𝑧𝑤))
1916, 18bitri 264 . . . . . . . . . . . . . . 15 (𝑤 ∈ (𝑥 ∖ {𝑧}) ↔ (𝑤𝑥𝑧𝑤))
2019anbi2i 729 . . . . . . . . . . . . . 14 (((𝑣𝑤𝑣𝑧) ∧ 𝑤 ∈ (𝑥 ∖ {𝑧})) ↔ ((𝑣𝑤𝑣𝑧) ∧ (𝑤𝑥𝑧𝑤)))
21 ancom 466 . . . . . . . . . . . . . . 15 ((𝑣𝑤𝑣𝑧) ↔ (𝑣𝑧𝑣𝑤))
2221anbi2ci 731 . . . . . . . . . . . . . 14 (((𝑣𝑤𝑣𝑧) ∧ (𝑤𝑥𝑧𝑤)) ↔ ((𝑤𝑥𝑧𝑤) ∧ (𝑣𝑧𝑣𝑤)))
23 anass 680 . . . . . . . . . . . . . 14 (((𝑤𝑥𝑧𝑤) ∧ (𝑣𝑧𝑣𝑤)) ↔ (𝑤𝑥 ∧ (𝑧𝑤 ∧ (𝑣𝑧𝑣𝑤))))
2420, 22, 233bitri 286 . . . . . . . . . . . . 13 (((𝑣𝑤𝑣𝑧) ∧ 𝑤 ∈ (𝑥 ∖ {𝑧})) ↔ (𝑤𝑥 ∧ (𝑧𝑤 ∧ (𝑣𝑧𝑣𝑤))))
25 an32 838 . . . . . . . . . . . . 13 (((𝑣𝑤𝑣𝑧) ∧ 𝑤 ∈ (𝑥 ∖ {𝑧})) ↔ ((𝑣𝑤𝑤 ∈ (𝑥 ∖ {𝑧})) ∧ 𝑣𝑧))
2624, 25bitr3i 266 . . . . . . . . . . . 12 ((𝑤𝑥 ∧ (𝑧𝑤 ∧ (𝑣𝑧𝑣𝑤))) ↔ ((𝑣𝑤𝑤 ∈ (𝑥 ∖ {𝑧})) ∧ 𝑣𝑧))
2715, 26bitr3i 266 . . . . . . . . . . 11 ((𝑤𝑥 ∧ ¬ (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤))) ↔ ((𝑣𝑤𝑤 ∈ (𝑥 ∖ {𝑧})) ∧ 𝑣𝑧))
2827exbii 1771 . . . . . . . . . 10 (∃𝑤(𝑤𝑥 ∧ ¬ (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤))) ↔ ∃𝑤((𝑣𝑤𝑤 ∈ (𝑥 ∖ {𝑧})) ∧ 𝑣𝑧))
29 19.41v 1911 . . . . . . . . . 10 (∃𝑤((𝑣𝑤𝑤 ∈ (𝑥 ∖ {𝑧})) ∧ 𝑣𝑧) ↔ (∃𝑤(𝑣𝑤𝑤 ∈ (𝑥 ∖ {𝑧})) ∧ 𝑣𝑧))
3010, 28, 293bitri 286 . . . . . . . . 9 (∃𝑤𝑥 ¬ (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)) ↔ (∃𝑤(𝑣𝑤𝑤 ∈ (𝑥 ∖ {𝑧})) ∧ 𝑣𝑧))
31 rexnal 2990 . . . . . . . . 9 (∃𝑤𝑥 ¬ (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)) ↔ ¬ ∀𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)))
329, 30, 313bitr2ri 289 . . . . . . . 8 (¬ ∀𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)) ↔ (𝑣 (𝑥 ∖ {𝑧}) ∧ 𝑣𝑧))
3332con1bii 346 . . . . . . 7 (¬ (𝑣 (𝑥 ∖ {𝑧}) ∧ 𝑣𝑧) ↔ ∀𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)))
347, 33bitr3i 266 . . . . . 6 ((¬ 𝑣 (𝑥 ∖ {𝑧}) ∨ ¬ 𝑣𝑧) ↔ ∀𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)))
3534a1i 11 . . . . 5 (𝑣𝑧 → ((¬ 𝑣 (𝑥 ∖ {𝑧}) ∨ ¬ 𝑣𝑧) ↔ ∀𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤))))
3635rabbiia 3176 . . . 4 {𝑣𝑧 ∣ (¬ 𝑣 (𝑥 ∖ {𝑧}) ∨ ¬ 𝑣𝑧)} = {𝑣𝑧 ∣ ∀𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤))}
371, 6, 363eqtri 2647 . . 3 (𝑧 (𝑥 ∖ {𝑧})) = {𝑣𝑧 ∣ ∀𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤))}
3837neeq1i 2854 . 2 ((𝑧 (𝑥 ∖ {𝑧})) ≠ ∅ ↔ {𝑣𝑧 ∣ ∀𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤))} ≠ ∅)
39 rabn0 3937 . 2 ({𝑣𝑧 ∣ ∀𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤))} ≠ ∅ ↔ ∃𝑣𝑧𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)))
4038, 39bitri 264 1 ((𝑧 (𝑥 ∖ {𝑧})) ≠ ∅ ↔ ∃𝑣𝑧𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  wex 1701  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  cdif 3556  cun 3557  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-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-nul 3897  df-sn 4154  df-uni 4408
This theorem is referenced by:  kmlem13  8936
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