Theorem kmlem11 10105

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 4883	. . . . 5 ⊢ ∪ 𝐴 = ∪ {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))}
3		vex 3450	. . . . . . 7 ⊢ 𝑡 ∈ V
4	3	difexi 5290	. . . . . 6 ⊢ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∈ V
5	4	dfiun2 4998	. . . . 5 ⊢ ∪ 𝑡 ∈ 𝑥 (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) = ∪ {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))}
6	2, 5	eqtr4i 2762	. . . 4 ⊢ ∪ 𝐴 = ∪ 𝑡 ∈ 𝑥 (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))
7	6	ineq2i 4174	. . 3 ⊢ (𝑧 ∩ ∪ 𝐴) = (𝑧 ∩ ∪ 𝑡 ∈ 𝑥 (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})))
8		iunin2 5036	. . 3 ⊢ ∪ 𝑡 ∈ 𝑥 (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = (𝑧 ∩ ∪ 𝑡 ∈ 𝑥 (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})))
9	7, 8	eqtr4i 2762	. 2 ⊢ (𝑧 ∩ ∪ 𝐴) = ∪ 𝑡 ∈ 𝑥 (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})))
10		undif2 4441	. . . . . 6 ⊢ ({𝑧} ∪ (𝑥 ∖ {𝑧})) = ({𝑧} ∪ 𝑥)
11		snssi 4773	. . . . . . 7 ⊢ (𝑧 ∈ 𝑥 → {𝑧} ⊆ 𝑥)
12		ssequn1 4145	. . . . . . 7 ⊢ ({𝑧} ⊆ 𝑥 ↔ ({𝑧} ∪ 𝑥) = 𝑥)
13	11, 12	sylib 217	. . . . . 6 ⊢ (𝑧 ∈ 𝑥 → ({𝑧} ∪ 𝑥) = 𝑥)
14	10, 13	eqtr2id 2784	. . . . 5 ⊢ (𝑧 ∈ 𝑥 → 𝑥 = ({𝑧} ∪ (𝑥 ∖ {𝑧})))
15	14	iuneq1d 4986	. . . 4 ⊢ (𝑧 ∈ 𝑥 → ∪ 𝑡 ∈ 𝑥 (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = ∪ 𝑡 ∈ ({𝑧} ∪ (𝑥 ∖ {𝑧}))(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))))
16		iunxun 5059	. . . . . 6 ⊢ ∪ 𝑡 ∈ ({𝑧} ∪ (𝑥 ∖ {𝑧}))(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = (∪ 𝑡 ∈ {𝑧} (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) ∪ ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))))
17		vex 3450	. . . . . . . 8 ⊢ 𝑧 ∈ V
18		difeq1 4080	. . . . . . . . . 10 ⊢ (𝑡 = 𝑧 → (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑡})))
19		sneq 4601	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑧 → {𝑡} = {𝑧})
20	19	difeq2d 4087	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑧 → (𝑥 ∖ {𝑡}) = (𝑥 ∖ {𝑧}))
21	20	unieqd 4884	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑧 → ∪ (𝑥 ∖ {𝑡}) = ∪ (𝑥 ∖ {𝑧}))
22	21	difeq2d 4087	. . . . . . . . . 10 ⊢ (𝑡 = 𝑧 → (𝑧 ∖ ∪ (𝑥 ∖ {𝑡})) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})))
23	18, 22	eqtrd 2771	. . . . . . . . 9 ⊢ (𝑡 = 𝑧 → (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})))
24	23	ineq2d 4177	. . . . . . . 8 ⊢ (𝑡 = 𝑧 → (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = (𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))))
25	17, 24	iunxsn 5056	. . . . . . 7 ⊢ ∪ 𝑡 ∈ {𝑧} (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = (𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})))
26	25	uneq1i 4124	. . . . . 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 2759	. . . . 5 ⊢ ∪ 𝑡 ∈ ({𝑧} ∪ (𝑥 ∖ {𝑧}))(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = ((𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))) ∪ ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))))
28		eldifsni 4755	. . . . . . . . . 10 ⊢ (𝑡 ∈ (𝑥 ∖ {𝑧}) → 𝑡 ≠ 𝑧)
29		incom 4166	. . . . . . . . . . . 12 ⊢ (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑧)
30		kmlem4 10098	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑡 ≠ 𝑧) → ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑧) = ∅)
31	29, 30	eqtrid 2783	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑡 ≠ 𝑧) → (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = ∅)
32	31	ex 413	. . . . . . . . . 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 3138	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑥 → ∀𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = ∅)
35		iuneq2 4978	. . . . . . . 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 5027	. . . . . . 7 ⊢ ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})∅ = ∅
38	36, 37	eqtrdi 2787	. . . . . 6 ⊢ (𝑧 ∈ 𝑥 → ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = ∅)
39	38	uneq2d 4128	. . . . 5 ⊢ (𝑧 ∈ 𝑥 → ((𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))) ∪ ∪ 𝑡 ∈ (𝑥 ∖ {𝑧})(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})))) = ((𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))) ∪ ∅))
40	27, 39	eqtrid 2783	. . . 4 ⊢ (𝑧 ∈ 𝑥 → ∪ 𝑡 ∈ ({𝑧} ∪ (𝑥 ∖ {𝑧}))(𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = ((𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))) ∪ ∅))
41	15, 40	eqtrd 2771	. . 3 ⊢ (𝑧 ∈ 𝑥 → ∪ 𝑡 ∈ 𝑥 (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = ((𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))) ∪ ∅))
42		un0 4355	. . . 4 ⊢ ((𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))) ∪ ∅) = (𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})))
43		indif 4234	. . . 4 ⊢ (𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))
44	42, 43	eqtri 2759	. . 3 ⊢ ((𝑧 ∩ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))) ∪ ∅) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))
45	41, 44	eqtrdi 2787	. 2 ⊢ (𝑧 ∈ 𝑥 → ∪ 𝑡 ∈ 𝑥 (𝑧 ∩ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})))
46	9, 45	eqtrid 2783	1 ⊢ (𝑧 ∈ 𝑥 → (𝑧 ∩ ∪ 𝐴) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2708   ≠ wne 2939  ∀wral 3060  ∃wrex 3069   ∖ cdif 3910   ∪ cun 3911   ∩ cin 3912   ⊆ wss 3913  ∅c0 4287  {csn 4591  ∪ cuni 4870  ∪ ciun 4959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2702  ax-sep 5261

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-sn 4592  df-uni 4871  df-iun 4961

This theorem is referenced by:  kmlem12  10106