Theorem kmlem11 10155

Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.)

Hypothesis

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

Assertion

Ref	Expression
kmlem11	⊢ (𝑧 ∈ 𝑥 → (𝑧 ∩ ∪ 𝐴) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})))

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

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

Proof of Theorem kmlem11

Step	Hyp	Ref	Expression
1		kmlem9.1	. . . . . 6 ⊢ 𝐴 = {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))}
2	1	unieqi 4922	. . . . 5 ⊢ ∪ 𝐴 = ∪ {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))}
3		vex 3479	. . . . . . 7 ⊢ 𝑡 ∈ V
4	3	difexi 5329	. . . . . 6 ⊢ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∈ V
5	4	dfiun2 5037	. . . . 5 ⊢ ∪ 𝑡 ∈ 𝑥 (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) = ∪ {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))}
6	2, 5	eqtr4i 2764	. . . 4 ⊢ ∪ 𝐴 = ∪ 𝑡 ∈ 𝑥 (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))
7	6	ineq2i 4210	. . 3 ⊢ (𝑧 ∩ ∪ 𝐴) = (𝑧 ∩ ∪ 𝑡 ∈ 𝑥 (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})))
8		iunin2 5075	. . 3 ⊢ ∪ 𝑡 ∈ 𝑥 (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = (𝑧 ∩ ∪ 𝑡 ∈ 𝑥 (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})))
9	7, 8	eqtr4i 2764	. 2 ⊢ (𝑧 ∩ ∪ 𝐴) = ∪ 𝑡 ∈ 𝑥 (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})))
10		undif2 4477	. . . . . 6 ⊢ ({𝑧} ∪ (𝑥 ∖ {𝑧})) = ({𝑧} ∪ 𝑥)
11		snssi 4812	. . . . . . 7 ⊢ (𝑧 ∈ 𝑥 → {𝑧} ⊆ 𝑥)
12		ssequn1 4181	. . . . . . 7 ⊢ ({𝑧} ⊆ 𝑥 ↔ ({𝑧} ∪ 𝑥) = 𝑥)
13	11, 12	sylib 217	. . . . . 6 ⊢ (𝑧 ∈ 𝑥 → ({𝑧} ∪ 𝑥) = 𝑥)
14	10, 13	eqtr2id 2786	. . . . 5 ⊢ (𝑧 ∈ 𝑥 → 𝑥 = ({𝑧} ∪ (𝑥 ∖ {𝑧})))
15	14	iuneq1d 5025	. . . 4 ⊢ (𝑧 ∈ 𝑥 → ∪ 𝑡 ∈ 𝑥 (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = ∪ 𝑡 ∈ ({𝑧} ∪ (𝑥 ∖ {𝑧}))(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))))
16		iunxun 5098	. . . . . 6 ⊢ ∪ 𝑡 ∈ ({𝑧} ∪ (𝑥 ∖ {𝑧}))(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = (∪ 𝑡 ∈ {𝑧} (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) ∪ ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))))
17		vex 3479	. . . . . . . 8 ⊢ 𝑧 ∈ V
18		difeq1 4116	. . . . . . . . . 10 ⊢ (𝑡 = 𝑧 → (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑡})))
19		sneq 4639	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑧 → {𝑡} = {𝑧})
20	19	difeq2d 4123	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑧 → (𝑥 ∖ {𝑡}) = (𝑥 ∖ {𝑧}))
21	20	unieqd 4923	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑧 → ∪ (𝑥 ∖ {𝑡}) = ∪ (𝑥 ∖ {𝑧}))
22	21	difeq2d 4123	. . . . . . . . . 10 ⊢ (𝑡 = 𝑧 → (𝑧 ∖ ∪ (𝑥 ∖ {𝑡})) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})))
23	18, 22	eqtrd 2773	. . . . . . . . 9 ⊢ (𝑡 = 𝑧 → (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})))
24	23	ineq2d 4213	. . . . . . . 8 ⊢ (𝑡 = 𝑧 → (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = (𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))))
25	17, 24	iunxsn 5095	. . . . . . 7 ⊢ ∪ 𝑡 ∈ {𝑧} (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = (𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})))
26	25	uneq1i 4160	. . . . . 6 ⊢ (∪ 𝑡 ∈ {𝑧} (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) ∪ ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})))) = ((𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))) ∪ ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))))
27	16, 26	eqtri 2761	. . . . 5 ⊢ ∪ 𝑡 ∈ ({𝑧} ∪ (𝑥 ∖ {𝑧}))(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = ((𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))) ∪ ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))))
28		eldifsni 4794	. . . . . . . . . 10 ⊢ (𝑡 ∈ (𝑥 ∖ {𝑧}) → 𝑡 ≠ 𝑧)
29		incom 4202	. . . . . . . . . . . 12 ⊢ (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑧)
30		kmlem4 10148	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑡 ≠ 𝑧) → ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑧) = ∅)
31	29, 30	eqtrid 2785	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑡 ≠ 𝑧) → (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = ∅)
32	31	ex 414	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑥 → (𝑡 ≠ 𝑧 → (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = ∅))
33	28, 32	syl5 34	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑥 → (𝑡 ∈ (𝑥 ∖ {𝑧}) → (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = ∅))
34	33	ralrimiv 3146	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑥 → ∀𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = ∅)
35		iuneq2 5017	. . . . . . . 8 ⊢ (∀𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = ∅ → ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})∅)
36	34, 35	syl 17	. . . . . . 7 ⊢ (𝑧 ∈ 𝑥 → ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})∅)
37		iun0 5066	. . . . . . 7 ⊢ ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})∅ = ∅
38	36, 37	eqtrdi 2789	. . . . . 6 ⊢ (𝑧 ∈ 𝑥 → ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = ∅)
39	38	uneq2d 4164	. . . . 5 ⊢ (𝑧 ∈ 𝑥 → ((𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))) ∪ ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})))) = ((𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))) ∪ ∅))
40	27, 39	eqtrid 2785	. . . 4 ⊢ (𝑧 ∈ 𝑥 → ∪ 𝑡 ∈ ({𝑧} ∪ (𝑥 ∖ {𝑧}))(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = ((𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))) ∪ ∅))
41	15, 40	eqtrd 2773	. . 3 ⊢ (𝑧 ∈ 𝑥 → ∪ 𝑡 ∈ 𝑥 (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = ((𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))) ∪ ∅))
42		un0 4391	. . . 4 ⊢ ((𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))) ∪ ∅) = (𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})))
43		indif 4270	. . . 4 ⊢ (𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))
44	42, 43	eqtri 2761	. . 3 ⊢ ((𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))) ∪ ∅) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))
45	41, 44	eqtrdi 2789	. 2 ⊢ (𝑧 ∈ 𝑥 → ∪ 𝑡 ∈ 𝑥 (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})))
46	9, 45	eqtrid 2785	1 ⊢ (𝑧 ∈ 𝑥 → (𝑧 ∩ ∪ 𝐴) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710   ≠ wne 2941  ∀wral 3062  ∃wrex 3071   ∖ cdif 3946   ∪ cun 3947   ∩ cin 3948   ⊆ wss 3949  ∅c0 4323  {csn 4629  ∪ cuni 4909  ∪ ciun 4998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-sep 5300

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-sn 4630  df-uni 4910  df-iun 5000

This theorem is referenced by:  kmlem12  10156