Theorem sepnsepo 49414

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 49413	. . . . 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 3162	. . 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 4183	. . . . . . 7 ⊢ (𝑧 ⊆ 𝑥 → (𝑧 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦))
7		sseq0 4344	. . . . . . . 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 3153	. . . 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 4160	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = 𝑧) → (𝑥 ∩ 𝑦) = (𝑧 ∩ 𝑦))
14	13	eqeq1d 2739	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = 𝑧) → ((𝑥 ∩ 𝑦) = ∅ ↔ (𝑧 ∩ 𝑦) = ∅))
15	14	rexbidv 3162	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = 𝑧) → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑧 ∩ 𝑦) = ∅))
16	2, 11, 15	opnneieqv 49401	. . 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 49412	. . . 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 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  ‘cfv 6493  Topctop 22871  neicnei 23075

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 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371

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 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-top 22872  df-nei 23076

This theorem is referenced by:  sepcsepo  49417  isnrm4  49421  iscnrm4  49444