Theorem ntreq0 23106

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 23065	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆))
3	2	eqeq1d 2742	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (((int‘𝐽)‘𝑆) = ∅ ↔ ∪ (𝐽 ∩ 𝒫 𝑆) = ∅))
4		neq0 4375	. . . . 5 ⊢ (¬ ∪ (𝐽 ∩ 𝒫 𝑆) = ∅ ↔ ∃𝑦 𝑦 ∈ ∪ (𝐽 ∩ 𝒫 𝑆))
5	4	con1bii 356	. . . 4 ⊢ (¬ ∃𝑦 𝑦 ∈ ∪ (𝐽 ∩ 𝒫 𝑆) ↔ ∪ (𝐽 ∩ 𝒫 𝑆) = ∅)
6		ancom 460	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 𝑆)) ↔ (𝑥 ∈ (𝐽 ∩ 𝒫 𝑆) ∧ 𝑦 ∈ 𝑥))
7		elin 3992	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐽 ∩ 𝒫 𝑆) ↔ (𝑥 ∈ 𝐽 ∧ 𝑥 ∈ 𝒫 𝑆))
8	7	anbi1i 623	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐽 ∩ 𝒫 𝑆) ∧ 𝑦 ∈ 𝑥) ↔ ((𝑥 ∈ 𝐽 ∧ 𝑥 ∈ 𝒫 𝑆) ∧ 𝑦 ∈ 𝑥))
9		anass 468	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑥 ∈ 𝒫 𝑆) ∧ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝐽 ∧ (𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥)))
10	6, 8, 9	3bitri 297	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 𝑆)) ↔ (𝑥 ∈ 𝐽 ∧ (𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥)))
11	10	exbii 1846	. . . . . . . 8 ⊢ (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 𝑆)) ↔ ∃𝑥(𝑥 ∈ 𝐽 ∧ (𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥)))
12		eluni 4934	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ (𝐽 ∩ 𝒫 𝑆) ↔ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 𝑆)))
13		df-rex 3077	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥) ↔ ∃𝑥(𝑥 ∈ 𝐽 ∧ (𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥)))
14	11, 12, 13	3bitr4i 303	. . . . . . 7 ⊢ (𝑦 ∈ ∪ (𝐽 ∩ 𝒫 𝑆) ↔ ∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥))
15	14	exbii 1846	. . . . . 6 ⊢ (∃𝑦 𝑦 ∈ ∪ (𝐽 ∩ 𝒫 𝑆) ↔ ∃𝑦∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥))
16		rexcom4 3294	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐽 ∃𝑦(𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥) ↔ ∃𝑦∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥))
17		19.42v 1953	. . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝒫 𝑆 ∧ ∃𝑦 𝑦 ∈ 𝑥))
18	17	rexbii 3100	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐽 ∃𝑦(𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥) ↔ ∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ ∃𝑦 𝑦 ∈ 𝑥))
19	15, 16, 18	3bitr2i 299	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ ∪ (𝐽 ∩ 𝒫 𝑆) ↔ ∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ ∃𝑦 𝑦 ∈ 𝑥))
20	19	notbii 320	. . . 4 ⊢ (¬ ∃𝑦 𝑦 ∈ ∪ (𝐽 ∩ 𝒫 𝑆) ↔ ¬ ∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ ∃𝑦 𝑦 ∈ 𝑥))
21	5, 20	bitr3i 277	. . 3 ⊢ (∪ (𝐽 ∩ 𝒫 𝑆) = ∅ ↔ ¬ ∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ ∃𝑦 𝑦 ∈ 𝑥))
22		ralinexa 3107	. . 3 ⊢ (∀𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 → ¬ ∃𝑦 𝑦 ∈ 𝑥) ↔ ¬ ∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ ∃𝑦 𝑦 ∈ 𝑥))
23		velpw 4627	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝑆 ↔ 𝑥 ⊆ 𝑆)
24		neq0 4375	. . . . . 6 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
25	24	con1bii 356	. . . . 5 ⊢ (¬ ∃𝑦 𝑦 ∈ 𝑥 ↔ 𝑥 = ∅)
26	23, 25	imbi12i 350	. . . 4 ⊢ ((𝑥 ∈ 𝒫 𝑆 → ¬ ∃𝑦 𝑦 ∈ 𝑥) ↔ (𝑥 ⊆ 𝑆 → 𝑥 = ∅))
27	26	ralbii 3099	. . 3 ⊢ (∀𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 → ¬ ∃𝑦 𝑦 ∈ 𝑥) ↔ ∀𝑥 ∈ 𝐽 (𝑥 ⊆ 𝑆 → 𝑥 = ∅))
28	21, 22, 27	3bitr2i 299	. 2 ⊢ (∪ (𝐽 ∩ 𝒫 𝑆) = ∅ ↔ ∀𝑥 ∈ 𝐽 (𝑥 ⊆ 𝑆 → 𝑥 = ∅))
29	3, 28	bitrdi 287	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (((int‘𝐽)‘𝑆) = ∅ ↔ ∀𝑥 ∈ 𝐽 (𝑥 ⊆ 𝑆 → 𝑥 = ∅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  ∪ cuni 4931  ‘cfv 6573  Topctop 22920  intcnt 23046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-top 22921  df-ntr 23049

This theorem is referenced by: (None)