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Theorem kmlem7 8929
Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 4 => 1. (Contributed by NM, 26-Mar-2004.)
Assertion
Ref Expression
kmlem7 ((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ¬ ∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
Distinct variable group:   𝑥,𝑣,𝑤,𝑧

Proof of Theorem kmlem7
StepHypRef Expression
1 kmlem6 8928 . 2 ((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∀𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)))
2 ralinexa 2992 . . . . . 6 (∀𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)) ↔ ¬ ∃𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
32rexbii 3035 . . . . 5 (∃𝑣𝑧𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)) ↔ ∃𝑣𝑧 ¬ ∃𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
4 rexnal 2990 . . . . 5 (∃𝑣𝑧 ¬ ∃𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ↔ ¬ ∀𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
53, 4bitri 264 . . . 4 (∃𝑣𝑧𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)) ↔ ¬ ∀𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
65ralbii 2975 . . 3 (∀𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)) ↔ ∀𝑧𝑥 ¬ ∀𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
7 ralnex 2987 . . 3 (∀𝑧𝑥 ¬ ∀𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ↔ ¬ ∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
86, 7bitri 264 . 2 (∀𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)) ↔ ¬ ∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
91, 8sylib 208 1 ((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ¬ ∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  cin 3558  c0 3896
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-v 3191  df-dif 3562  df-nul 3897
This theorem is referenced by:  kmlem13  8935
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