Theorem sepnsepo 49761

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 23	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝐽 ∈ Top)
3	2	sepnsepolem2 49760	. . . . 5 ⊢ (𝐽 ∈ Top → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))
4	3	anbi2d 642	. . . 4 ⊢ (𝐽 ∈ Top → ((𝐶 ⊆ 𝑥 ∧ ∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅) ↔ (𝐶 ⊆ 𝑥 ∧ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))))
5	4	rexbidv 3191	. . 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 4194	. . . . . . 7 ⊢ (𝑧 ⊆ 𝑥 → (𝑧 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦))
7		sseq0 4361	. . . . . . . 8 ⊢ (((𝑧 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑧 ∩ 𝑦) = ∅)
8	7	ex 418	. . . . . . 7 ⊢ ((𝑧 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦) → ((𝑥 ∩ 𝑦) = ∅ → (𝑧 ∩ 𝑦) = ∅))
9	6, 8	syl 18	. . . . . 6 ⊢ (𝑧 ⊆ 𝑥 → ((𝑥 ∩ 𝑦) = ∅ → (𝑧 ∩ 𝑦) = ∅))
10	9	adantl 487	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ⊆ 𝑥) → ((𝑥 ∩ 𝑦) = ∅ → (𝑧 ∩ 𝑦) = ∅))
11	10	reximdv 3182	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ⊆ 𝑥) → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ → ∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑧 ∩ 𝑦) = ∅))
12		simpr 490	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = 𝑧) → 𝑥 = 𝑧)
13	12	ineq1d 4172	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = 𝑧) → (𝑥 ∩ 𝑦) = (𝑧 ∩ 𝑦))
14	13	eqeq1d 2767	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = 𝑧) → ((𝑥 ∩ 𝑦) = ∅ ↔ (𝑧 ∩ 𝑦) = ∅))
15	14	rexbidv 3191	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = 𝑧) → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑧 ∩ 𝑦) = ∅))
16	2, 11, 15	opnneieqv 49748	. . 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 49759	. . . 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 18	1 ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝐶)∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∃wrex 3091   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  ‘cfv 6540  Topctop 23102  neicnei 23306

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-top 23103  df-nei 23307

This theorem is used by:  sepcsepo  49764  isnrm4  49768  iscnrm4  49791