Theorem ntreq0 21684

Description: Two ways to say that a subset has an empty interior. (Contributed by NM, 3-Oct-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

Ref	Expression
clscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
ntreq0	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (((int‘𝐽)‘𝑆) = ∅ ↔ ∀𝑥 ∈ 𝐽 (𝑥 ⊆ 𝑆 → 𝑥 = ∅)))

Distinct variable groups:   𝑥,𝐽   𝑥,𝑆   𝑥,𝑋

Proof of Theorem ntreq0

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clscld.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2	1	ntrval 21643	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆))
3	2	eqeq1d 2823	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (((int‘𝐽)‘𝑆) = ∅ ↔ ∪ (𝐽 ∩ 𝒫 𝑆) = ∅))
4		neq0 4308	. . . . 5 ⊢ (¬ ∪ (𝐽 ∩ 𝒫 𝑆) = ∅ ↔ ∃𝑦 𝑦 ∈ ∪ (𝐽 ∩ 𝒫 𝑆))
5	4	con1bii 359	. . . 4 ⊢ (¬ ∃𝑦 𝑦 ∈ ∪ (𝐽 ∩ 𝒫 𝑆) ↔ ∪ (𝐽 ∩ 𝒫 𝑆) = ∅)
6		ancom 463	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 𝑆)) ↔ (𝑥 ∈ (𝐽 ∩ 𝒫 𝑆) ∧ 𝑦 ∈ 𝑥))
7		elin 4168	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐽 ∩ 𝒫 𝑆) ↔ (𝑥 ∈ 𝐽 ∧ 𝑥 ∈ 𝒫 𝑆))
8	7	anbi1i 625	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐽 ∩ 𝒫 𝑆) ∧ 𝑦 ∈ 𝑥) ↔ ((𝑥 ∈ 𝐽 ∧ 𝑥 ∈ 𝒫 𝑆) ∧ 𝑦 ∈ 𝑥))
9		anass 471	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑥 ∈ 𝒫 𝑆) ∧ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝐽 ∧ (𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥)))
10	6, 8, 9	3bitri 299	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 𝑆)) ↔ (𝑥 ∈ 𝐽 ∧ (𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥)))
11	10	exbii 1844	. . . . . . . 8 ⊢ (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 𝑆)) ↔ ∃𝑥(𝑥 ∈ 𝐽 ∧ (𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥)))
12		eluni 4840	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ (𝐽 ∩ 𝒫 𝑆) ↔ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 𝑆)))
13		df-rex 3144	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥) ↔ ∃𝑥(𝑥 ∈ 𝐽 ∧ (𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥)))
14	11, 12, 13	3bitr4i 305	. . . . . . 7 ⊢ (𝑦 ∈ ∪ (𝐽 ∩ 𝒫 𝑆) ↔ ∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥))
15	14	exbii 1844	. . . . . 6 ⊢ (∃𝑦 𝑦 ∈ ∪ (𝐽 ∩ 𝒫 𝑆) ↔ ∃𝑦∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥))
16		rexcom4 3249	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐽 ∃𝑦(𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥) ↔ ∃𝑦∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥))
17		19.42v 1950	. . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝒫 𝑆 ∧ ∃𝑦 𝑦 ∈ 𝑥))
18	17	rexbii 3247	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐽 ∃𝑦(𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥) ↔ ∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ ∃𝑦 𝑦 ∈ 𝑥))
19	15, 16, 18	3bitr2i 301	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ ∪ (𝐽 ∩ 𝒫 𝑆) ↔ ∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ ∃𝑦 𝑦 ∈ 𝑥))
20	19	notbii 322	. . . 4 ⊢ (¬ ∃𝑦 𝑦 ∈ ∪ (𝐽 ∩ 𝒫 𝑆) ↔ ¬ ∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ ∃𝑦 𝑦 ∈ 𝑥))
21	5, 20	bitr3i 279	. . 3 ⊢ (∪ (𝐽 ∩ 𝒫 𝑆) = ∅ ↔ ¬ ∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ ∃𝑦 𝑦 ∈ 𝑥))
22		ralinexa 3264	. . 3 ⊢ (∀𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 → ¬ ∃𝑦 𝑦 ∈ 𝑥) ↔ ¬ ∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ ∃𝑦 𝑦 ∈ 𝑥))
23		velpw 4543	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝑆 ↔ 𝑥 ⊆ 𝑆)
24		neq0 4308	. . . . . 6 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
25	24	con1bii 359	. . . . 5 ⊢ (¬ ∃𝑦 𝑦 ∈ 𝑥 ↔ 𝑥 = ∅)
26	23, 25	imbi12i 353	. . . 4 ⊢ ((𝑥 ∈ 𝒫 𝑆 → ¬ ∃𝑦 𝑦 ∈ 𝑥) ↔ (𝑥 ⊆ 𝑆 → 𝑥 = ∅))
27	26	ralbii 3165	. . 3 ⊢ (∀𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 → ¬ ∃𝑦 𝑦 ∈ 𝑥) ↔ ∀𝑥 ∈ 𝐽 (𝑥 ⊆ 𝑆 → 𝑥 = ∅))
28	21, 22, 27	3bitr2i 301	. 2 ⊢ (∪ (𝐽 ∩ 𝒫 𝑆) = ∅ ↔ ∀𝑥 ∈ 𝐽 (𝑥 ⊆ 𝑆 → 𝑥 = ∅))
29	3, 28	syl6bb 289	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (((int‘𝐽)‘𝑆) = ∅ ↔ ∀𝑥 ∈ 𝐽 (𝑥 ⊆ 𝑆 → 𝑥 = ∅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139   ∩ cin 3934   ⊆ wss 3935  ∅c0 4290  𝒫 cpw 4538  ∪ cuni 4837  ‘cfv 6354  Topctop 21500  intcnt 21624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-top 21501  df-ntr 21627

This theorem is referenced by: (None)