Theorem kmlem6 10237

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 3122	. 2 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ ∧ ∀𝑤 ∈ 𝑥 (𝜑 → 𝐴 = ∅)) ↔ (∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝜑 → 𝐴 = ∅)))
2		n0 4299	. . . . 5 ⊢ (𝑧 ≠ ∅ ↔ ∃𝑣 𝑣 ∈ 𝑧)
3	2	biimpi 219	. . . 4 ⊢ (𝑧 ≠ ∅ → ∃𝑣 𝑣 ∈ 𝑧)
4		ne0i 4286	. . . . . . . 8 ⊢ (𝑣 ∈ 𝐴 → 𝐴 ≠ ∅)
5	4	necon2bi 2985	. . . . . . 7 ⊢ (𝐴 = ∅ → ¬ 𝑣 ∈ 𝐴)
6	5	imim2i 17	. . . . . 6 ⊢ ((𝜑 → 𝐴 = ∅) → (𝜑 → ¬ 𝑣 ∈ 𝐴))
7	6	ralimi 3099	. . . . 5 ⊢ (∀𝑤 ∈ 𝑥 (𝜑 → 𝐴 = ∅) → ∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴))
8	7	alrimiv 1959	. . . 4 ⊢ (∀𝑤 ∈ 𝑥 (𝜑 → 𝐴 = ∅) → ∀𝑣∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴))
9		19.29r 1906	. . . . 5 ⊢ ((∃𝑣 𝑣 ∈ 𝑧 ∧ ∀𝑣∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴)) → ∃𝑣(𝑣 ∈ 𝑧 ∧ ∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴)))
10		df-rex 3087	. . . . 5 ⊢ (∃𝑣 ∈ 𝑧 ∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴) ↔ ∃𝑣(𝑣 ∈ 𝑧 ∧ ∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴)))
11	9, 10	sylibr 237	. . . 4 ⊢ ((∃𝑣 𝑣 ∈ 𝑧 ∧ ∀𝑣∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴)) → ∃𝑣 ∈ 𝑧 ∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴))
12	3, 8, 11	syl2an 608	. . 3 ⊢ ((𝑧 ≠ ∅ ∧ ∀𝑤 ∈ 𝑥 (𝜑 → 𝐴 = ∅)) → ∃𝑣 ∈ 𝑧 ∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴))
13	12	ralimi 3099	. 2 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ ∧ ∀𝑤 ∈ 𝑥 (𝜑 → 𝐴 = ∅)) → ∀𝑧 ∈ 𝑥 ∃𝑣 ∈ 𝑧 ∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴))
14	1, 13	sylbir 238	1 ⊢ ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝜑 → 𝐴 = ∅)) → ∀𝑧 ∈ 𝑥 ∃𝑣 ∈ 𝑧 ∀𝑤 ∈ 𝑥 (𝜑 → ¬ 𝑣 ∈ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  ∅c0 4278

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-dif 4047  df-nul 4279

This theorem is used by:  kmlem7  10238