Theorem ntreq0 23021

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 22980	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆))
3	2	eqeq1d 2738	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (((int‘𝐽)‘𝑆) = ∅ ↔ ∪ (𝐽 ∩ 𝒫 𝑆) = ∅))
4		neq0 4304	. . . . 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 3917	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐽 ∩ 𝒫 𝑆) ↔ (𝑥 ∈ 𝐽 ∧ 𝑥 ∈ 𝒫 𝑆))
8	7	anbi1i 624	. . . . . . . . . 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 1849	. . . . . . . 8 ⊢ (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 𝑆)) ↔ ∃𝑥(𝑥 ∈ 𝐽 ∧ (𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥)))
12		eluni 4866	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ (𝐽 ∩ 𝒫 𝑆) ↔ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 𝑆)))
13		df-rex 3061	. . . . . . . 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 1849	. . . . . 6 ⊢ (∃𝑦 𝑦 ∈ ∪ (𝐽 ∩ 𝒫 𝑆) ↔ ∃𝑦∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥))
16		rexcom4 3263	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐽 ∃𝑦(𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥) ↔ ∃𝑦∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥))
17		19.42v 1954	. . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝒫 𝑆 ∧ ∃𝑦 𝑦 ∈ 𝑥))
18	17	rexbii 3083	. . . . . 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 3089	. . 3 ⊢ (∀𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 → ¬ ∃𝑦 𝑦 ∈ 𝑥) ↔ ¬ ∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ ∃𝑦 𝑦 ∈ 𝑥))
23		velpw 4559	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝑆 ↔ 𝑥 ⊆ 𝑆)
24		neq0 4304	. . . . . 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 3082	. . 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 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  ∪ cuni 4863  ‘cfv 6492  Topctop 22837  intcnt 22961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-top 22838  df-ntr 22964

This theorem is referenced by: (None)