Theorem opnssneib 23089

Description: Any superset of an open set is a neighborhood of it. (Contributed by NM, 14-Feb-2007.)

Hypothesis

Ref	Expression
neips.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
opnssneib	⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑆 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑆)))

Proof of Theorem opnssneib

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 769	. . . . . 6 ⊢ (((𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ 𝑁) → 𝑁 ⊆ 𝑋)
2		sseq2 3949	. . . . . . . . . 10 ⊢ (𝑔 = 𝑆 → (𝑆 ⊆ 𝑔 ↔ 𝑆 ⊆ 𝑆))
3		sseq1 3948	. . . . . . . . . 10 ⊢ (𝑔 = 𝑆 → (𝑔 ⊆ 𝑁 ↔ 𝑆 ⊆ 𝑁))
4	2, 3	anbi12d 633	. . . . . . . . 9 ⊢ (𝑔 = 𝑆 → ((𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁) ↔ (𝑆 ⊆ 𝑆 ∧ 𝑆 ⊆ 𝑁)))
5		ssid 3945	. . . . . . . . . 10 ⊢ 𝑆 ⊆ 𝑆
6	5	biantrur 530	. . . . . . . . 9 ⊢ (𝑆 ⊆ 𝑁 ↔ (𝑆 ⊆ 𝑆 ∧ 𝑆 ⊆ 𝑁))
7	4, 6	bitr4di 289	. . . . . . . 8 ⊢ (𝑔 = 𝑆 → ((𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁) ↔ 𝑆 ⊆ 𝑁))
8	7	rspcev 3565	. . . . . . 7 ⊢ ((𝑆 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑁) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))
9	8	adantlr 716	. . . . . 6 ⊢ (((𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ 𝑁) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))
10	1, 9	jca 511	. . . . 5 ⊢ (((𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ 𝑁) → (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)))
11	10	ex 412	. . . 4 ⊢ ((𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑆 ⊆ 𝑁 → (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
12	11	3adant1 1131	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑆 ⊆ 𝑁 → (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
13		neips.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
14	13	eltopss 22881	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ 𝑋)
15	13	isnei 23077	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
16	14, 15	syldan 592	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
17	16	3adant3 1133	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
18	12, 17	sylibrd 259	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑆 ⊆ 𝑁 → 𝑁 ∈ ((nei‘𝐽)‘𝑆)))
19		ssnei 23084	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑁)
20	19	ex 412	. . 3 ⊢ (𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝑆 ⊆ 𝑁))
21	20	3ad2ant1 1134	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝑆 ⊆ 𝑁))
22	18, 21	impbid 212	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑆 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑆)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ⊆ wss 3890  ∪ cuni 4851  ‘cfv 6490  Topctop 22867  neicnei 23071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-top 22868  df-nei 23072

This theorem is referenced by:  neissex  23101