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Theorem kmlem7 8861
 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 8860 . 2 ((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∀𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)))
2 ralinexa 2980 . . . . . 6 (∀𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)) ↔ ¬ ∃𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
32rexbii 3023 . . . . 5 (∃𝑣𝑧𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)) ↔ ∃𝑣𝑧 ¬ ∃𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
4 rexnal 2978 . . . . 5 (∃𝑣𝑧 ¬ ∃𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ↔ ¬ ∀𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
53, 4bitri 263 . . . 4 (∃𝑣𝑧𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)) ↔ ¬ ∀𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
65ralbii 2963 . . 3 (∀𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)) ↔ ∀𝑧𝑥 ¬ ∀𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
7 ralnex 2975 . . 3 (∀𝑧𝑥 ¬ ∀𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ↔ ¬ ∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
86, 7bitri 263 . 2 (∀𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)) ↔ ¬ ∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
91, 8sylib 207 1 ((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ¬ ∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∩ cin 3539  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-nul 3875 This theorem is referenced by:  kmlem13  8867
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