Theorem nconnsubb 23480

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 23477	. . . 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 593	. . 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 1141	. . . 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 4165	. . . . . . . . 9 ⊢ (𝑥 = 𝑈 → (𝑥 ∩ 𝐴) = (𝑈 ∩ 𝐴))
13	12	neeq1d 3016	. . . . . . . 8 ⊢ (𝑥 = 𝑈 → ((𝑥 ∩ 𝐴) ≠ ∅ ↔ (𝑈 ∩ 𝐴) ≠ ∅))
14		ineq1 4165	. . . . . . . . . 10 ⊢ (𝑥 = 𝑈 → (𝑥 ∩ 𝑦) = (𝑈 ∩ 𝑦))
15	14	ineq1d 4171	. . . . . . . . 9 ⊢ (𝑥 = 𝑈 → ((𝑥 ∩ 𝑦) ∩ 𝐴) = ((𝑈 ∩ 𝑦) ∩ 𝐴))
16	15	eqeq1d 2764	. . . . . . . 8 ⊢ (𝑥 = 𝑈 → (((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅ ↔ ((𝑈 ∩ 𝑦) ∩ 𝐴) = ∅))
17	13, 16	3anbi13d 1459	. . . . . . 7 ⊢ (𝑥 = 𝑈 → (((𝑥 ∩ 𝐴) ≠ ∅ ∧ (𝑦 ∩ 𝐴) ≠ ∅ ∧ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅) ↔ ((𝑈 ∩ 𝐴) ≠ ∅ ∧ (𝑦 ∩ 𝐴) ≠ ∅ ∧ ((𝑈 ∩ 𝑦) ∩ 𝐴) = ∅)))
18		uneq1 4114	. . . . . . . . 9 ⊢ (𝑥 = 𝑈 → (𝑥 ∪ 𝑦) = (𝑈 ∪ 𝑦))
19	18	ineq1d 4171	. . . . . . . 8 ⊢ (𝑥 = 𝑈 → ((𝑥 ∪ 𝑦) ∩ 𝐴) = ((𝑈 ∪ 𝑦) ∩ 𝐴))
20	19	neeq1d 3016	. . . . . . 7 ⊢ (𝑥 = 𝑈 → (((𝑥 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴 ↔ ((𝑈 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴))
21	17, 20	imbi12d 346	. . . . . 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 4165	. . . . . . . . 9 ⊢ (𝑦 = 𝑉 → (𝑦 ∩ 𝐴) = (𝑉 ∩ 𝐴))
23	22	neeq1d 3016	. . . . . . . 8 ⊢ (𝑦 = 𝑉 → ((𝑦 ∩ 𝐴) ≠ ∅ ↔ (𝑉 ∩ 𝐴) ≠ ∅))
24		ineq2 4166	. . . . . . . . . 10 ⊢ (𝑦 = 𝑉 → (𝑈 ∩ 𝑦) = (𝑈 ∩ 𝑉))
25	24	ineq1d 4171	. . . . . . . . 9 ⊢ (𝑦 = 𝑉 → ((𝑈 ∩ 𝑦) ∩ 𝐴) = ((𝑈 ∩ 𝑉) ∩ 𝐴))
26	25	eqeq1d 2764	. . . . . . . 8 ⊢ (𝑦 = 𝑉 → (((𝑈 ∩ 𝑦) ∩ 𝐴) = ∅ ↔ ((𝑈 ∩ 𝑉) ∩ 𝐴) = ∅))
27	23, 26	3anbi23d 1460	. . . . . . 7 ⊢ (𝑦 = 𝑉 → (((𝑈 ∩ 𝐴) ≠ ∅ ∧ (𝑦 ∩ 𝐴) ≠ ∅ ∧ ((𝑈 ∩ 𝑦) ∩ 𝐴) = ∅) ↔ ((𝑈 ∩ 𝐴) ≠ ∅ ∧ (𝑉 ∩ 𝐴) ≠ ∅ ∧ ((𝑈 ∩ 𝑉) ∩ 𝐴) = ∅)))
28		sseqin2 4175	. . . . . . . . 9 ⊢ (𝐴 ⊆ (𝑈 ∪ 𝑦) ↔ ((𝑈 ∪ 𝑦) ∩ 𝐴) = 𝐴)
29	28	necon3bbii 3004	. . . . . . . 8 ⊢ (¬ 𝐴 ⊆ (𝑈 ∪ 𝑦) ↔ ((𝑈 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴)
30		uneq2 4115	. . . . . . . . . 10 ⊢ (𝑦 = 𝑉 → (𝑈 ∪ 𝑦) = (𝑈 ∪ 𝑉))
31	30	sseq2d 3968	. . . . . . . . 9 ⊢ (𝑦 = 𝑉 → (𝐴 ⊆ (𝑈 ∪ 𝑦) ↔ 𝐴 ⊆ (𝑈 ∪ 𝑉)))
32	31	notbid 320	. . . . . . . 8 ⊢ (𝑦 = 𝑉 → (¬ 𝐴 ⊆ (𝑈 ∪ 𝑦) ↔ ¬ 𝐴 ⊆ (𝑈 ∪ 𝑉)))
33	29, 32	bitr3id 287	. . . . . . 7 ⊢ (𝑦 = 𝑉 → (((𝑈 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴 ↔ ¬ 𝐴 ⊆ (𝑈 ∪ 𝑉)))
34	27, 33	imbi12d 346	. . . . . 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 3592	. . . . 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 593	. . . 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 242	. 2 ⊢ (𝜑 → ((𝐽 ↾t 𝐴) ∈ Conn → ¬ 𝐴 ⊆ (𝑈 ∪ 𝑉)))
39	1, 38	mt2d 136	1 ⊢ (𝜑 → ¬ (𝐽 ↾t 𝐴) ∈ Conn)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  ∀wral 3076   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  ‘cfv 6521  (class class class)co 7396   ↾_t crest 17449  TopOnctopon 22967  Conncconn 23468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-en 8928  df-fin 8931  df-fi 9357  df-rest 17451  df-topgen 17472  df-top 22951  df-topon 22968  df-bases 23003  df-cld 23076  df-conn 23469

This theorem is referenced by:  iunconnlem  23484  clsconn  23487  reconnlem1  24884  ordtconnlem1  34218