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Theorem sepnsepo 48522
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 48521 . . . . 5 (𝐽 ∈ Top → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥𝑦) = ∅ ↔ ∃𝑦𝐽 (𝐷𝑦 ∧ (𝑥𝑦) = ∅)))
43anbi2d 629 . . . 4 (𝐽 ∈ Top → ((𝐶𝑥 ∧ ∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥𝑦) = ∅) ↔ (𝐶𝑥 ∧ ∃𝑦𝐽 (𝐷𝑦 ∧ (𝑥𝑦) = ∅))))
54rexbidv 3181 . . 3 (𝐽 ∈ Top → (∃𝑥𝐽 (𝐶𝑥 ∧ ∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥𝑦) = ∅) ↔ ∃𝑥𝐽 (𝐶𝑥 ∧ ∃𝑦𝐽 (𝐷𝑦 ∧ (𝑥𝑦) = ∅))))
6 ssrin 4257 . . . . . . 7 (𝑧𝑥 → (𝑧𝑦) ⊆ (𝑥𝑦))
7 sseq0 4422 . . . . . . . 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 3172 . . . 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 4234 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 = 𝑧) → (𝑥𝑦) = (𝑧𝑦))
1413eqeq1d 2736 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 = 𝑧) → ((𝑥𝑦) = ∅ ↔ (𝑧𝑦) = ∅))
1514rexbidv 3181 . . . 4 ((𝐽 ∈ Top ∧ 𝑥 = 𝑧) → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥𝑦) = ∅ ↔ ∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑧𝑦) = ∅))
162, 11, 15opnneieqv 48509 . . 3 (𝐽 ∈ Top → (∃𝑥 ∈ ((nei‘𝐽)‘𝐶)∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥𝑦) = ∅ ↔ ∃𝑥𝐽 (𝐶𝑥 ∧ ∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥𝑦) = ∅)))
17 sepnsepolem1 48520 . . . 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 206  wa 395  w3a 1087   = wceq 1537  wcel 2103  wrex 3072  cin 3969  wss 3970  c0 4347  cfv 6572  Topctop 22913  neicnei 23119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-top 22914  df-nei 23120
This theorem is referenced by:  sepcsepo  48525  isnrm4  48529  iscnrm4  48553
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