Theorem nconnsubb 21226

Description: Disconnectedness for a subspace. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)

Hypotheses

Ref	Expression
nconnsubb.2	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
nconnsubb.3	⊢ (𝜑 → 𝐴 ⊆ 𝑋)
nconnsubb.4	⊢ (𝜑 → 𝑈 ∈ 𝐽)
nconnsubb.5	⊢ (𝜑 → 𝑉 ∈ 𝐽)
nconnsubb.6	⊢ (𝜑 → (𝑈 ∩ 𝐴) ≠ ∅)
nconnsubb.7	⊢ (𝜑 → (𝑉 ∩ 𝐴) ≠ ∅)
nconnsubb.8	⊢ (𝜑 → ((𝑈 ∩ 𝑉) ∩ 𝐴) = ∅)
nconnsubb.9	⊢ (𝜑 → 𝐴 ⊆ (𝑈 ∪ 𝑉))

Assertion

Ref	Expression
nconnsubb	⊢ (𝜑 → ¬ (𝐽 ↾t 𝐴) ∈ Conn)

Proof of Theorem nconnsubb

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nconnsubb.9	. 2 ⊢ (𝜑 → 𝐴 ⊆ (𝑈 ∪ 𝑉))
2		nconnsubb.2	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
3		nconnsubb.3	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
4		connsuba 21223	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((𝐽 ↾t 𝐴) ∈ Conn ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (((𝑥 ∩ 𝐴) ≠ ∅ ∧ (𝑦 ∩ 𝐴) ≠ ∅ ∧ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅) → ((𝑥 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴)))
5	2, 3, 4	syl2anc 693	. . 3 ⊢ (𝜑 → ((𝐽 ↾t 𝐴) ∈ Conn ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (((𝑥 ∩ 𝐴) ≠ ∅ ∧ (𝑦 ∩ 𝐴) ≠ ∅ ∧ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅) → ((𝑥 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴)))
6		nconnsubb.6	. . . . 5 ⊢ (𝜑 → (𝑈 ∩ 𝐴) ≠ ∅)
7		nconnsubb.7	. . . . 5 ⊢ (𝜑 → (𝑉 ∩ 𝐴) ≠ ∅)
8		nconnsubb.8	. . . . 5 ⊢ (𝜑 → ((𝑈 ∩ 𝑉) ∩ 𝐴) = ∅)
9	6, 7, 8	3jca 1242	. . . 4 ⊢ (𝜑 → ((𝑈 ∩ 𝐴) ≠ ∅ ∧ (𝑉 ∩ 𝐴) ≠ ∅ ∧ ((𝑈 ∩ 𝑉) ∩ 𝐴) = ∅))
10		nconnsubb.4	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝐽)
11		nconnsubb.5	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝐽)
12		ineq1 3807	. . . . . . . . 9 ⊢ (𝑥 = 𝑈 → (𝑥 ∩ 𝐴) = (𝑈 ∩ 𝐴))
13	12	neeq1d 2853	. . . . . . . 8 ⊢ (𝑥 = 𝑈 → ((𝑥 ∩ 𝐴) ≠ ∅ ↔ (𝑈 ∩ 𝐴) ≠ ∅))
14		ineq1 3807	. . . . . . . . . 10 ⊢ (𝑥 = 𝑈 → (𝑥 ∩ 𝑦) = (𝑈 ∩ 𝑦))
15	14	ineq1d 3813	. . . . . . . . 9 ⊢ (𝑥 = 𝑈 → ((𝑥 ∩ 𝑦) ∩ 𝐴) = ((𝑈 ∩ 𝑦) ∩ 𝐴))
16	15	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑥 = 𝑈 → (((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅ ↔ ((𝑈 ∩ 𝑦) ∩ 𝐴) = ∅))
17	13, 16	3anbi13d 1401	. . . . . . 7 ⊢ (𝑥 = 𝑈 → (((𝑥 ∩ 𝐴) ≠ ∅ ∧ (𝑦 ∩ 𝐴) ≠ ∅ ∧ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅) ↔ ((𝑈 ∩ 𝐴) ≠ ∅ ∧ (𝑦 ∩ 𝐴) ≠ ∅ ∧ ((𝑈 ∩ 𝑦) ∩ 𝐴) = ∅)))
18		uneq1 3760	. . . . . . . . 9 ⊢ (𝑥 = 𝑈 → (𝑥 ∪ 𝑦) = (𝑈 ∪ 𝑦))
19	18	ineq1d 3813	. . . . . . . 8 ⊢ (𝑥 = 𝑈 → ((𝑥 ∪ 𝑦) ∩ 𝐴) = ((𝑈 ∪ 𝑦) ∩ 𝐴))
20	19	neeq1d 2853	. . . . . . 7 ⊢ (𝑥 = 𝑈 → (((𝑥 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴 ↔ ((𝑈 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴))
21	17, 20	imbi12d 334	. . . . . 6 ⊢ (𝑥 = 𝑈 → ((((𝑥 ∩ 𝐴) ≠ ∅ ∧ (𝑦 ∩ 𝐴) ≠ ∅ ∧ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅) → ((𝑥 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴) ↔ (((𝑈 ∩ 𝐴) ≠ ∅ ∧ (𝑦 ∩ 𝐴) ≠ ∅ ∧ ((𝑈 ∩ 𝑦) ∩ 𝐴) = ∅) → ((𝑈 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴)))
22		ineq1 3807	. . . . . . . . 9 ⊢ (𝑦 = 𝑉 → (𝑦 ∩ 𝐴) = (𝑉 ∩ 𝐴))
23	22	neeq1d 2853	. . . . . . . 8 ⊢ (𝑦 = 𝑉 → ((𝑦 ∩ 𝐴) ≠ ∅ ↔ (𝑉 ∩ 𝐴) ≠ ∅))
24		ineq2 3808	. . . . . . . . . 10 ⊢ (𝑦 = 𝑉 → (𝑈 ∩ 𝑦) = (𝑈 ∩ 𝑉))
25	24	ineq1d 3813	. . . . . . . . 9 ⊢ (𝑦 = 𝑉 → ((𝑈 ∩ 𝑦) ∩ 𝐴) = ((𝑈 ∩ 𝑉) ∩ 𝐴))
26	25	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑦 = 𝑉 → (((𝑈 ∩ 𝑦) ∩ 𝐴) = ∅ ↔ ((𝑈 ∩ 𝑉) ∩ 𝐴) = ∅))
27	23, 26	3anbi23d 1402	. . . . . . 7 ⊢ (𝑦 = 𝑉 → (((𝑈 ∩ 𝐴) ≠ ∅ ∧ (𝑦 ∩ 𝐴) ≠ ∅ ∧ ((𝑈 ∩ 𝑦) ∩ 𝐴) = ∅) ↔ ((𝑈 ∩ 𝐴) ≠ ∅ ∧ (𝑉 ∩ 𝐴) ≠ ∅ ∧ ((𝑈 ∩ 𝑉) ∩ 𝐴) = ∅)))
28		sseqin2 3817	. . . . . . . . 9 ⊢ (𝐴 ⊆ (𝑈 ∪ 𝑦) ↔ ((𝑈 ∪ 𝑦) ∩ 𝐴) = 𝐴)
29	28	necon3bbii 2841	. . . . . . . 8 ⊢ (¬ 𝐴 ⊆ (𝑈 ∪ 𝑦) ↔ ((𝑈 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴)
30		uneq2 3761	. . . . . . . . . 10 ⊢ (𝑦 = 𝑉 → (𝑈 ∪ 𝑦) = (𝑈 ∪ 𝑉))
31	30	sseq2d 3633	. . . . . . . . 9 ⊢ (𝑦 = 𝑉 → (𝐴 ⊆ (𝑈 ∪ 𝑦) ↔ 𝐴 ⊆ (𝑈 ∪ 𝑉)))
32	31	notbid 308	. . . . . . . 8 ⊢ (𝑦 = 𝑉 → (¬ 𝐴 ⊆ (𝑈 ∪ 𝑦) ↔ ¬ 𝐴 ⊆ (𝑈 ∪ 𝑉)))
33	29, 32	syl5bbr 274	. . . . . . 7 ⊢ (𝑦 = 𝑉 → (((𝑈 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴 ↔ ¬ 𝐴 ⊆ (𝑈 ∪ 𝑉)))
34	27, 33	imbi12d 334	. . . . . 6 ⊢ (𝑦 = 𝑉 → ((((𝑈 ∩ 𝐴) ≠ ∅ ∧ (𝑦 ∩ 𝐴) ≠ ∅ ∧ ((𝑈 ∩ 𝑦) ∩ 𝐴) = ∅) → ((𝑈 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴) ↔ (((𝑈 ∩ 𝐴) ≠ ∅ ∧ (𝑉 ∩ 𝐴) ≠ ∅ ∧ ((𝑈 ∩ 𝑉) ∩ 𝐴) = ∅) → ¬ 𝐴 ⊆ (𝑈 ∪ 𝑉))))
35	21, 34	rspc2v 3322	. . . . 5 ⊢ ((𝑈 ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) → (∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (((𝑥 ∩ 𝐴) ≠ ∅ ∧ (𝑦 ∩ 𝐴) ≠ ∅ ∧ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅) → ((𝑥 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴) → (((𝑈 ∩ 𝐴) ≠ ∅ ∧ (𝑉 ∩ 𝐴) ≠ ∅ ∧ ((𝑈 ∩ 𝑉) ∩ 𝐴) = ∅) → ¬ 𝐴 ⊆ (𝑈 ∪ 𝑉))))
36	10, 11, 35	syl2anc 693	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (((𝑥 ∩ 𝐴) ≠ ∅ ∧ (𝑦 ∩ 𝐴) ≠ ∅ ∧ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅) → ((𝑥 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴) → (((𝑈 ∩ 𝐴) ≠ ∅ ∧ (𝑉 ∩ 𝐴) ≠ ∅ ∧ ((𝑈 ∩ 𝑉) ∩ 𝐴) = ∅) → ¬ 𝐴 ⊆ (𝑈 ∪ 𝑉))))
37	9, 36	mpid 44	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (((𝑥 ∩ 𝐴) ≠ ∅ ∧ (𝑦 ∩ 𝐴) ≠ ∅ ∧ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅) → ((𝑥 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴) → ¬ 𝐴 ⊆ (𝑈 ∪ 𝑉)))
38	5, 37	sylbid 230	. 2 ⊢ (𝜑 → ((𝐽 ↾t 𝐴) ∈ Conn → ¬ 𝐴 ⊆ (𝑈 ∪ 𝑉)))
39	1, 38	mt2d 131	1 ⊢ (𝜑 → ¬ (𝐽 ↾t 𝐴) ∈ Conn)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  ‘cfv 5888  (class class class)co 6650   ↾_t crest 16081  TopOnctopon 20715  Conncconn 21214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-conn 21215

This theorem is referenced by:  iunconnlem  21230  clsconn  21233  reconnlem1  22629  ordtconnlem1  29970