Theorem kmlem10 10162

Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)

Hypothesis

Ref	Expression
kmlem9.1	⊢ 𝐴 = {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))}

Assertion

Ref	Expression
kmlem10	⊢ (∀ℎ(∀𝑧 ∈ ℎ ∀𝑤 ∈ ℎ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∃𝑦∀𝑧 ∈ ℎ 𝜑) → ∃𝑦∀𝑧 ∈ 𝐴 𝜑)

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑢,𝑡,ℎ   𝑦,𝐴,𝑧,𝑤,ℎ   𝜑,ℎ

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

Proof of Theorem kmlem10

Step	Hyp	Ref	Expression
1		kmlem9.1	. . 3 ⊢ 𝐴 = {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))}
2	1	kmlem9 10161	. 2 ⊢ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)
3		vex 3462	. . . . 5 ⊢ 𝑥 ∈ V
4	3	abrexex 7968	. . . 4 ⊢ {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))} ∈ V
5	1, 4	eqeltri 2862	. . 3 ⊢ 𝐴 ∈ V
6		raleq 3323	. . . . 5 ⊢ (ℎ = 𝐴 → (∀𝑤 ∈ ℎ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) ↔ ∀𝑤 ∈ 𝐴 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)))
7	6	raleqbi1dv 3336	. . . 4 ⊢ (ℎ = 𝐴 → (∀𝑧 ∈ ℎ ∀𝑤 ∈ ℎ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)))
8		raleq 3323	. . . . 5 ⊢ (ℎ = 𝐴 → (∀𝑧 ∈ ℎ 𝜑 ↔ ∀𝑧 ∈ 𝐴 𝜑))
9	8	exbidv 1954	. . . 4 ⊢ (ℎ = 𝐴 → (∃𝑦∀𝑧 ∈ ℎ 𝜑 ↔ ∃𝑦∀𝑧 ∈ 𝐴 𝜑))
10	7, 9	imbi12d 347	. . 3 ⊢ (ℎ = 𝐴 → ((∀𝑧 ∈ ℎ ∀𝑤 ∈ ℎ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∃𝑦∀𝑧 ∈ ℎ 𝜑) ↔ (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∃𝑦∀𝑧 ∈ 𝐴 𝜑)))
11	5, 10	spcv 3567	. 2 ⊢ (∀ℎ(∀𝑧 ∈ ℎ ∀𝑤 ∈ ℎ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∃𝑦∀𝑧 ∈ ℎ 𝜑) → (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∃𝑦∀𝑧 ∈ 𝐴 𝜑))
12	2, 11	mpi 21	1 ⊢ (∀ℎ(∀𝑧 ∈ ℎ ∀𝑤 ∈ ℎ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∃𝑦∀𝑧 ∈ ℎ 𝜑) → ∃𝑦∀𝑧 ∈ 𝐴 𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568   = wceq 1570  ∃wex 1812  {cab 2744   ≠ wne 2961  ∀wral 3082  ∃wrex 3092  Vcvv 3458   ∖ cdif 3905   ∩ cin 3907  ∅c0 4289  {csn 4594  ∪ cuni 4877

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-rep 5243

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2570  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-in 3915  df-ss 3925  df-nul 4290  df-sn 4595  df-uni 4878

This theorem is used by:  kmlem13  10165