Theorem ntreq0 22971

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 22930	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆))
3	2	eqeq1d 2732	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (((int‘𝐽)‘𝑆) = ∅ ↔ ∪ (𝐽 ∩ 𝒫 𝑆) = ∅))
4		neq0 4318	. . . . 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 3933	. . . . . . . . . . 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 1848	. . . . . . . 8 ⊢ (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 𝑆)) ↔ ∃𝑥(𝑥 ∈ 𝐽 ∧ (𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥)))
12		eluni 4877	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ (𝐽 ∩ 𝒫 𝑆) ↔ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 𝑆)))
13		df-rex 3055	. . . . . . . 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 1848	. . . . . 6 ⊢ (∃𝑦 𝑦 ∈ ∪ (𝐽 ∩ 𝒫 𝑆) ↔ ∃𝑦∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥))
16		rexcom4 3265	. . . . . 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 3077	. . . . . 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 3084	. . 3 ⊢ (∀𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 → ¬ ∃𝑦 𝑦 ∈ 𝑥) ↔ ¬ ∃𝑥 ∈ 𝐽 (𝑥 ∈ 𝒫 𝑆 ∧ ∃𝑦 𝑦 ∈ 𝑥))
23		velpw 4571	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝑆 ↔ 𝑥 ⊆ 𝑆)
24		neq0 4318	. . . . . 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 3076	. . 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 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299  𝒫 cpw 4566  ∪ cuni 4874  ‘cfv 6514  Topctop 22787  intcnt 22911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-top 22788  df-ntr 22914

This theorem is referenced by: (None)