Theorem sepnsepo 48885

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 48884	. . . . 5 ⊢ (𝐽 ∈ Top → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))
4	3	anbi2d 630	. . . 4 ⊢ (𝐽 ∈ Top → ((𝐶 ⊆ 𝑥 ∧ ∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅) ↔ (𝐶 ⊆ 𝑥 ∧ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))))
5	4	rexbidv 3157	. . 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 4201	. . . . . . 7 ⊢ (𝑧 ⊆ 𝑥 → (𝑧 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦))
7		sseq0 4362	. . . . . . . 8 ⊢ (((𝑧 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑧 ∩ 𝑦) = ∅)
8	7	ex 412	. . . . . . 7 ⊢ ((𝑧 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦) → ((𝑥 ∩ 𝑦) = ∅ → (𝑧 ∩ 𝑦) = ∅))
9	6, 8	syl 17	. . . . . 6 ⊢ (𝑧 ⊆ 𝑥 → ((𝑥 ∩ 𝑦) = ∅ → (𝑧 ∩ 𝑦) = ∅))
10	9	adantl 481	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ⊆ 𝑥) → ((𝑥 ∩ 𝑦) = ∅ → (𝑧 ∩ 𝑦) = ∅))
11	10	reximdv 3148	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ⊆ 𝑥) → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ → ∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑧 ∩ 𝑦) = ∅))
12		simpr 484	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = 𝑧) → 𝑥 = 𝑧)
13	12	ineq1d 4178	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = 𝑧) → (𝑥 ∩ 𝑦) = (𝑧 ∩ 𝑦))
14	13	eqeq1d 2731	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = 𝑧) → ((𝑥 ∩ 𝑦) = ∅ ↔ (𝑧 ∩ 𝑦) = ∅))
15	14	rexbidv 3157	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = 𝑧) → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑧 ∩ 𝑦) = ∅))
16	2, 11, 15	opnneieqv 48872	. . 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 48883	. . . 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 311	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  ‘cfv 6499  Topctop 22756  neicnei 22960

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-top 22757  df-nei 22961

This theorem is referenced by:  sepcsepo  48888  isnrm4  48892  iscnrm4  48915