Theorem kmlem6 10109

Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 4 => 1. (Contributed by NM, 26-Mar-2004.)

Assertion

Ref	Expression
kmlem6	⊢ ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝜑 → 𝐴 = ∅)) → ∀𝑧 ∈ 𝑥 ∃𝑣 ∈ 𝑧 ∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴))

Distinct variable groups:   𝑣,𝐴   𝑥,𝑣,𝜑   𝑤,𝑣,𝑧,𝑥

Allowed substitution hints:   𝜑(𝑧,𝑤)   𝐴(𝑥,𝑧,𝑤)

Proof of Theorem kmlem6

Step	Hyp	Ref	Expression
1		r19.26 3121	. 2 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ ∧ ∀𝑤 ∈ 𝑥 (𝜑 → 𝐴 = ∅)) ↔ (∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝜑 → 𝐴 = ∅)))
2		n0 4305	. . . . 5 ⊢ (𝑧 ≠ ∅ ↔ ∃𝑣 𝑣 ∈ 𝑧)
3	2	biimpi 218	. . . 4 ⊢ (𝑧 ≠ ∅ → ∃𝑣 𝑣 ∈ 𝑧)
4		ne0i 4293	. . . . . . . 8 ⊢ (𝑣 ∈ 𝐴 → 𝐴 ≠ ∅)
5	4	necon2bi 2986	. . . . . . 7 ⊢ (𝐴 = ∅ → ¬ 𝑣 ∈ 𝐴)
6	5	imim2i 16	. . . . . 6 ⊢ ((𝜑 → 𝐴 = ∅) → (𝜑 → ¬ 𝑣 ∈ 𝐴))
7	6	ralimi 3098	. . . . 5 ⊢ (∀𝑤 ∈ 𝑥 (𝜑 → 𝐴 = ∅) → ∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴))
8	7	alrimiv 1946	. . . 4 ⊢ (∀𝑤 ∈ 𝑥 (𝜑 → 𝐴 = ∅) → ∀𝑣∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴))
9		19.29r 1893	. . . . 5 ⊢ ((∃𝑣 𝑣 ∈ 𝑧 ∧ ∀𝑣∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴)) → ∃𝑣(𝑣 ∈ 𝑧 ∧ ∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴)))
10		df-rex 3086	. . . . 5 ⊢ (∃𝑣 ∈ 𝑧 ∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴) ↔ ∃𝑣(𝑣 ∈ 𝑧 ∧ ∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴)))
11	9, 10	sylibr 236	. . . 4 ⊢ ((∃𝑣 𝑣 ∈ 𝑧 ∧ ∀𝑣∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴)) → ∃𝑣 ∈ 𝑧 ∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴))
12	3, 8, 11	syl2an 605	. . 3 ⊢ ((𝑧 ≠ ∅ ∧ ∀𝑤 ∈ 𝑥 (𝜑 → 𝐴 = ∅)) → ∃𝑣 ∈ 𝑧 ∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴))
13	12	ralimi 3098	. 2 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ ∧ ∀𝑤 ∈ 𝑥 (𝜑 → 𝐴 = ∅)) → ∀𝑧 ∈ 𝑥 ∃𝑣 ∈ 𝑧 ∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴))
14	1, 13	sylbir 237	1 ⊢ ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝜑 → 𝐴 = ∅)) → ∀𝑧 ∈ 𝑥 ∃𝑣 ∈ 𝑧 ∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1557   = wceq 1559  ∃wex 1798   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  ∃wrex 3085  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-dif 3907  df-nul 4286

This theorem is referenced by:  kmlem7  10110