Theorem sepnsepo 45656

Description: Open neighborhood and neighborhood is equivalent regarding disjointness for both sides. Namely, separatedness by open neighborhoods is equivalent to separatedness by neighborhoods. (Contributed by Zhi Wang, 1-Sep-2024.)

Hypothesis

Ref	Expression
sepnsepolem2.1	⊢ (𝜑 → 𝐽 ∈ Top)

Assertion

Ref	Expression
sepnsepo	⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝐶)∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))

Distinct variable groups:   𝑦,𝐷   𝑦,𝐽,𝑥   𝑥,𝐶,𝑦   𝑥,𝐷   𝑥,𝐽

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sepnsepo

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sepnsepolem2.1	. 2 ⊢ (𝜑 → 𝐽 ∈ Top)
2		id 22	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝐽 ∈ Top)
3	2	sepnsepolem2 45655	. . . . 5 ⊢ (𝐽 ∈ Top → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))
4	3	anbi2d 631	. . . 4 ⊢ (𝐽 ∈ Top → ((𝐶 ⊆ 𝑥 ∧ ∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅) ↔ (𝐶 ⊆ 𝑥 ∧ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))))
5	4	rexbidv 3221	. . 3 ⊢ (𝐽 ∈ Top → (∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ ∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅) ↔ ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))))
6		ssrin 4140	. . . . . . 7 ⊢ (𝑧 ⊆ 𝑥 → (𝑧 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦))
7		sseq0 4298	. . . . . . . 8 ⊢ (((𝑧 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑧 ∩ 𝑦) = ∅)
8	7	ex 416	. . . . . . 7 ⊢ ((𝑧 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦) → ((𝑥 ∩ 𝑦) = ∅ → (𝑧 ∩ 𝑦) = ∅))
9	6, 8	syl 17	. . . . . 6 ⊢ (𝑧 ⊆ 𝑥 → ((𝑥 ∩ 𝑦) = ∅ → (𝑧 ∩ 𝑦) = ∅))
10	9	adantl 485	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ⊆ 𝑥) → ((𝑥 ∩ 𝑦) = ∅ → (𝑧 ∩ 𝑦) = ∅))
11	10	reximdv 3197	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ⊆ 𝑥) → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ → ∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑧 ∩ 𝑦) = ∅))
12		simpr 488	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = 𝑧) → 𝑥 = 𝑧)
13	12	ineq1d 4118	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = 𝑧) → (𝑥 ∩ 𝑦) = (𝑧 ∩ 𝑦))
14	13	eqeq1d 2760	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = 𝑧) → ((𝑥 ∩ 𝑦) = ∅ ↔ (𝑧 ∩ 𝑦) = ∅))
15	14	rexbidv 3221	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = 𝑧) → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑧 ∩ 𝑦) = ∅))
16	2, 11, 15	opnneieqv 45643	. . 3 ⊢ (𝐽 ∈ Top → (∃𝑥 ∈ ((nei‘𝐽)‘𝐶)∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ ∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅)))
17		sepnsepolem1 45654	. . . 4 ⊢ (∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅) ↔ ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))
18	17	a1i 11	. . 3 ⊢ (𝐽 ∈ Top → (∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅) ↔ ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))))
19	5, 16, 18	3bitr4d 314	. 2 ⊢ (𝐽 ∈ Top → (∃𝑥 ∈ ((nei‘𝐽)‘𝐶)∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))
20	1, 19	syl 17	1 ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝐶)∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃wrex 3071   ∩ cin 3859   ⊆ wss 3860  ∅c0 4227  ‘cfv 6340  Topctop 21606  neicnei 21810

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-top 21607  df-nei 21811

This theorem is referenced by:  sepcsepo  45659  isnrm4  45663  iscnrm4  45687