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Theorem sepnsepo 47768
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 (𝜑𝐽 ∈ Top)
Assertion
Ref Expression
sepnsepo (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝐶)∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥𝑦) = ∅ ↔ ∃𝑥𝐽𝑦𝐽 (𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅)))
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.
StepHypRef Expression
1 sepnsepolem2.1 . 2 (𝜑𝐽 ∈ Top)
2 id 22 . . . . . 6 (𝐽 ∈ Top → 𝐽 ∈ Top)
32sepnsepolem2 47767 . . . . 5 (𝐽 ∈ Top → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥𝑦) = ∅ ↔ ∃𝑦𝐽 (𝐷𝑦 ∧ (𝑥𝑦) = ∅)))
43anbi2d 628 . . . 4 (𝐽 ∈ Top → ((𝐶𝑥 ∧ ∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥𝑦) = ∅) ↔ (𝐶𝑥 ∧ ∃𝑦𝐽 (𝐷𝑦 ∧ (𝑥𝑦) = ∅))))
54rexbidv 3170 . . 3 (𝐽 ∈ Top → (∃𝑥𝐽 (𝐶𝑥 ∧ ∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥𝑦) = ∅) ↔ ∃𝑥𝐽 (𝐶𝑥 ∧ ∃𝑦𝐽 (𝐷𝑦 ∧ (𝑥𝑦) = ∅))))
6 ssrin 4226 . . . . . . 7 (𝑧𝑥 → (𝑧𝑦) ⊆ (𝑥𝑦))
7 sseq0 4392 . . . . . . . 8 (((𝑧𝑦) ⊆ (𝑥𝑦) ∧ (𝑥𝑦) = ∅) → (𝑧𝑦) = ∅)
87ex 412 . . . . . . 7 ((𝑧𝑦) ⊆ (𝑥𝑦) → ((𝑥𝑦) = ∅ → (𝑧𝑦) = ∅))
96, 8syl 17 . . . . . 6 (𝑧𝑥 → ((𝑥𝑦) = ∅ → (𝑧𝑦) = ∅))
109adantl 481 . . . . 5 ((𝐽 ∈ Top ∧ 𝑧𝑥) → ((𝑥𝑦) = ∅ → (𝑧𝑦) = ∅))
1110reximdv 3162 . . . 4 ((𝐽 ∈ Top ∧ 𝑧𝑥) → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥𝑦) = ∅ → ∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑧𝑦) = ∅))
12 simpr 484 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑥 = 𝑧) → 𝑥 = 𝑧)
1312ineq1d 4204 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 = 𝑧) → (𝑥𝑦) = (𝑧𝑦))
1413eqeq1d 2726 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 = 𝑧) → ((𝑥𝑦) = ∅ ↔ (𝑧𝑦) = ∅))
1514rexbidv 3170 . . . 4 ((𝐽 ∈ Top ∧ 𝑥 = 𝑧) → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥𝑦) = ∅ ↔ ∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑧𝑦) = ∅))
162, 11, 15opnneieqv 47755 . . 3 (𝐽 ∈ Top → (∃𝑥 ∈ ((nei‘𝐽)‘𝐶)∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥𝑦) = ∅ ↔ ∃𝑥𝐽 (𝐶𝑥 ∧ ∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥𝑦) = ∅)))
17 sepnsepolem1 47766 . . . 4 (∃𝑥𝐽𝑦𝐽 (𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅) ↔ ∃𝑥𝐽 (𝐶𝑥 ∧ ∃𝑦𝐽 (𝐷𝑦 ∧ (𝑥𝑦) = ∅)))
1817a1i 11 . . 3 (𝐽 ∈ Top → (∃𝑥𝐽𝑦𝐽 (𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅) ↔ ∃𝑥𝐽 (𝐶𝑥 ∧ ∃𝑦𝐽 (𝐷𝑦 ∧ (𝑥𝑦) = ∅))))
195, 16, 183bitr4d 311 . 2 (𝐽 ∈ Top → (∃𝑥 ∈ ((nei‘𝐽)‘𝐶)∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥𝑦) = ∅ ↔ ∃𝑥𝐽𝑦𝐽 (𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅)))
201, 19syl 17 1 (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝐶)∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥𝑦) = ∅ ↔ ∃𝑥𝐽𝑦𝐽 (𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wrex 3062  cin 3940  wss 3941  c0 4315  cfv 6534  Topctop 22719  neicnei 22925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-top 22720  df-nei 22926
This theorem is referenced by:  sepcsepo  47771  isnrm4  47775  iscnrm4  47799
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