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Theorem kmlem7 10197
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 10196 . 2 ((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∀𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)))
2 ralinexa 3101 . . . . . 6 (∀𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)) ↔ ¬ ∃𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
32rexbii 3094 . . . . 5 (∃𝑣𝑧𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)) ↔ ∃𝑣𝑧 ¬ ∃𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
4 rexnal 3100 . . . . 5 (∃𝑣𝑧 ¬ ∃𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ↔ ¬ ∀𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
53, 4bitri 275 . . . 4 (∃𝑣𝑧𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)) ↔ ¬ ∀𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
65ralbii 3093 . . 3 (∀𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)) ↔ ∀𝑧𝑥 ¬ ∀𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
7 ralnex 3072 . . 3 (∀𝑧𝑥 ¬ ∀𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ↔ ¬ ∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
86, 7bitri 275 . 2 (∀𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)) ↔ ¬ ∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
91, 8sylib 218 1 ((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ¬ ∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  cin 3950  c0 4333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-dif 3954  df-nul 4334
This theorem is referenced by:  kmlem13  10203
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