Theorem kmlem6 3585

Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 4 => 1.

Assertion

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

Distinct variable group(s):   x,z,w,v   φ,v   v,A

Proof of Theorem kmlem6

Step	Hyp	Ref	Expression
1		r19.26 1289	. 2 ⊢ (∀z ∈ x (¬ z = ∅ ∧ ∀w ∈ x (φ → A = ∅)) ↔ (∀z ∈ x ¬ z = ∅ ∧ ∀z ∈ x ∀w ∈ x (φ → A = ∅)))
2		19.29r 753	. . . . 5 ⊢ ((∃v v ∈ z ∧ ∀v∀w ∈ x (φ → ¬ v ∈ A)) → ∃v(v ∈ z ∧ ∀w ∈ x (φ → ¬ v ∈ A)))
3		df-rex 1206	. . . . 5 ⊢ (∃v ∈ z ∀w ∈ x (φ → ¬ v ∈ A) ↔ ∃v(v ∈ z ∧ ∀w ∈ x (φ → ¬ v ∈ A)))
4	2, 3	sylibr 175	. . . 4 ⊢ ((∃v v ∈ z ∧ ∀v∀w ∈ x (φ → ¬ v ∈ A)) → ∃v ∈ z ∀w ∈ x (φ → ¬ v ∈ A))
5		n0 1714	. . . . 5 ⊢ (¬ z = ∅ ↔ ∃v v ∈ z)
6	5	biimp 133	. . . 4 ⊢ (¬ z = ∅ → ∃v v ∈ z)
7		n0i 1712	. . . . . . . 8 ⊢ (v ∈ A → ¬ A = ∅)
8	7	con2i 89	. . . . . . 7 ⊢ (A = ∅ → ¬ v ∈ A)
9	8	syl3 18	. . . . . 6 ⊢ ((φ → A = ∅) → (φ → ¬ v ∈ A))
10	9	r19.20si 1254	. . . . 5 ⊢ (∀w ∈ x (φ → A = ∅) → ∀w ∈ x (φ → ¬ v ∈ A))
11	10	19.21aiv 943	. . . 4 ⊢ (∀w ∈ x (φ → A = ∅) → ∀v∀w ∈ x (φ → ¬ v ∈ A))
12	4, 6, 11	syl2an 349	. . 3 ⊢ ((¬ z = ∅ ∧ ∀w ∈ x (φ → A = ∅)) → ∃v ∈ z ∀w ∈ x (φ → ¬ v ∈ A))
13	12	r19.20si 1254	. 2 ⊢ (∀z ∈ x (¬ z = ∅ ∧ ∀w ∈ x (φ → A = ∅)) → ∀z ∈ x ∃v ∈ z ∀w ∈ x (φ → ¬ v ∈ A))
14	1, 13	sylbir 176	1 ⊢ ((∀z ∈ x ¬ z = ∅ ∧ ∀z ∈ x ∀w ∈ x (φ → A = ∅)) → ∀z ∈ x ∃v ∈ z ∀w ∈ x (φ → ¬ v ∈ A))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  ∅c0 1707

This theorem is referenced by:  kmlem7 3586

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-nul 1708

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