Theorem kmlem9 10241

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
kmlem9	⊢ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)

Distinct variable groups:   𝑥,𝑧,𝑤,𝑢,𝑡   𝑧,𝐴,𝑤

Allowed substitution hints:   𝐴(𝑥, 𝑢, 𝑡)

Proof of Theorem kmlem9

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3454	. . . 4 ⊢ 𝑧 ∈ V
2		eqeq1 2764	. . . . 5 ⊢ (𝑢 = 𝑧 → (𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ↔ 𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))))
3	2	rexbidv 3186	. . . 4 ⊢ (𝑢 = 𝑧 → (∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ↔ ∃𝑡 ∈ 𝑥 𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))))
4		kmlem9.1	. . . 4 ⊢ 𝐴 = {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))}
5	1, 3, 4	elab2 3635	. . 3 ⊢ (𝑧 ∈ 𝐴 ↔ ∃𝑡 ∈ 𝑥 𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})))
6		vex 3454	. . . . 5 ⊢ 𝑤 ∈ V
7		eqeq1 2764	. . . . . 6 ⊢ (𝑢 = 𝑤 → (𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ↔ 𝑤 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))))
8	7	rexbidv 3186	. . . . 5 ⊢ (𝑢 = 𝑤 → (∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ↔ ∃𝑡 ∈ 𝑥 𝑤 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))))
9	6, 8, 4	elab2 3635	. . . 4 ⊢ (𝑤 ∈ 𝐴 ↔ ∃𝑡 ∈ 𝑥 𝑤 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})))
10		difeq1 4056	. . . . . . 7 ⊢ (𝑡 = ℎ → (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) = (ℎ ∖ ∪ (𝑥 ∖ {𝑡})))
11		sneq 4590	. . . . . . . . . 10 ⊢ (𝑡 = ℎ → {𝑡} = {ℎ})
12	11	difeq2d 4063	. . . . . . . . 9 ⊢ (𝑡 = ℎ → (𝑥 ∖ {𝑡}) = (𝑥 ∖ {ℎ}))
13	12	unieqd 4879	. . . . . . . 8 ⊢ (𝑡 = ℎ → ∪ (𝑥 ∖ {𝑡}) = ∪ (𝑥 ∖ {ℎ}))
14	13	difeq2d 4063	. . . . . . 7 ⊢ (𝑡 = ℎ → (ℎ ∖ ∪ (𝑥 ∖ {𝑡})) = (ℎ ∖ ∪ (𝑥 ∖ {ℎ})))
15	10, 14	eqtrd 2795	. . . . . 6 ⊢ (𝑡 = ℎ → (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) = (ℎ ∖ ∪ (𝑥 ∖ {ℎ})))
16	15	eqeq2d 2771	. . . . 5 ⊢ (𝑡 = ℎ → (𝑤 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ↔ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))))
17	16	cbvrexvw 3241	. . . 4 ⊢ (∃𝑡 ∈ 𝑥 𝑤 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ↔ ∃ℎ ∈ 𝑥 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ})))
18	9, 17	bitri 278	. . 3 ⊢ (𝑤 ∈ 𝐴 ↔ ∃ℎ ∈ 𝑥 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ})))
19		reeanv 3234	. . . 4 ⊢ (∃𝑡 ∈ 𝑥 ∃ℎ ∈ 𝑥 (𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) ↔ (∃𝑡 ∈ 𝑥 𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ ∃ℎ ∈ 𝑥 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))))
20		eqeq12 2777	. . . . . . . . . 10 ⊢ ((𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → (𝑧 = 𝑤 ↔ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))))
21	15, 20	imbitrrid 249	. . . . . . . . 9 ⊢ ((𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → (𝑡 = ℎ → 𝑧 = 𝑤))
22	21	necon3d 2976	. . . . . . . 8 ⊢ ((𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → (𝑧 ≠ 𝑤 → 𝑡 ≠ ℎ))
23		kmlem5 10237	. . . . . . . . . 10 ⊢ ((ℎ ∈ 𝑥 ∧ 𝑡 ≠ ℎ) → ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) = ∅)
24		ineq12 4112	. . . . . . . . . . 11 ⊢ ((𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → (𝑧 ∩ 𝑤) = ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))))
25	24	eqeq1d 2762	. . . . . . . . . 10 ⊢ ((𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → ((𝑧 ∩ 𝑤) = ∅ ↔ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) = ∅))
26	23, 25	imbitrrid 249	. . . . . . . . 9 ⊢ ((𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → ((ℎ ∈ 𝑥 ∧ 𝑡 ≠ ℎ) → (𝑧 ∩ 𝑤) = ∅))
27	26	expd 421	. . . . . . . 8 ⊢ ((𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → (ℎ ∈ 𝑥 → (𝑡 ≠ ℎ → (𝑧 ∩ 𝑤) = ∅)))
28	22, 27	syl5d 74	. . . . . . 7 ⊢ ((𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → (ℎ ∈ 𝑥 → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)))
29	28	com12 33	. . . . . 6 ⊢ (ℎ ∈ 𝑥 → ((𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)))
30	29	adantl 487	. . . . 5 ⊢ ((𝑡 ∈ 𝑥 ∧ ℎ ∈ 𝑥) → ((𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)))
31	30	rexlimivv 3204	. . . 4 ⊢ (∃𝑡 ∈ 𝑥 ∃ℎ ∈ 𝑥 (𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))
32	19, 31	sylbir 238	. . 3 ⊢ ((∃𝑡 ∈ 𝑥 𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ ∃ℎ ∈ 𝑥 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))
33	5, 18, 32	syl2anb 610	. 2 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))
34	33	rgen2 3202	1 ⊢ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086   ∖ cdif 4045   ∩ cin 4089  ∅c0 4278  {csn 4584  ∪ cuni 4866

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-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-nul 4279  df-sn 4585  df-uni 4867

This theorem is used by:  kmlem10  10242