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Theorem sepnsepo 47830
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 (πœ‘ β†’ 𝐽 ∈ Top)
Assertion
Ref Expression
sepnsepo (πœ‘ β†’ (βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜πΆ)βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π·)(π‘₯ ∩ 𝑦) = βˆ… ↔ βˆƒπ‘₯ ∈ 𝐽 βˆƒπ‘¦ ∈ 𝐽 (𝐢 βŠ† π‘₯ ∧ 𝐷 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…)))
Distinct variable groups:   𝑦,𝐷   𝑦,𝐽,π‘₯   π‘₯,𝐢,𝑦   π‘₯,𝐷   π‘₯,𝐽
Allowed substitution hints:   πœ‘(π‘₯,𝑦)

Proof of Theorem sepnsepo
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sepnsepolem2.1 . 2 (πœ‘ β†’ 𝐽 ∈ Top)
2 id 22 . . . . . 6 (𝐽 ∈ Top β†’ 𝐽 ∈ Top)
32sepnsepolem2 47829 . . . . 5 (𝐽 ∈ Top β†’ (βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π·)(π‘₯ ∩ 𝑦) = βˆ… ↔ βˆƒπ‘¦ ∈ 𝐽 (𝐷 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…)))
43anbi2d 628 . . . 4 (𝐽 ∈ Top β†’ ((𝐢 βŠ† π‘₯ ∧ βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π·)(π‘₯ ∩ 𝑦) = βˆ…) ↔ (𝐢 βŠ† π‘₯ ∧ βˆƒπ‘¦ ∈ 𝐽 (𝐷 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…))))
54rexbidv 3172 . . 3 (𝐽 ∈ Top β†’ (βˆƒπ‘₯ ∈ 𝐽 (𝐢 βŠ† π‘₯ ∧ βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π·)(π‘₯ ∩ 𝑦) = βˆ…) ↔ βˆƒπ‘₯ ∈ 𝐽 (𝐢 βŠ† π‘₯ ∧ βˆƒπ‘¦ ∈ 𝐽 (𝐷 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…))))
6 ssrin 4228 . . . . . . 7 (𝑧 βŠ† π‘₯ β†’ (𝑧 ∩ 𝑦) βŠ† (π‘₯ ∩ 𝑦))
7 sseq0 4394 . . . . . . . 8 (((𝑧 ∩ 𝑦) βŠ† (π‘₯ ∩ 𝑦) ∧ (π‘₯ ∩ 𝑦) = βˆ…) β†’ (𝑧 ∩ 𝑦) = βˆ…)
87ex 412 . . . . . . 7 ((𝑧 ∩ 𝑦) βŠ† (π‘₯ ∩ 𝑦) β†’ ((π‘₯ ∩ 𝑦) = βˆ… β†’ (𝑧 ∩ 𝑦) = βˆ…))
96, 8syl 17 . . . . . 6 (𝑧 βŠ† π‘₯ β†’ ((π‘₯ ∩ 𝑦) = βˆ… β†’ (𝑧 ∩ 𝑦) = βˆ…))
109adantl 481 . . . . 5 ((𝐽 ∈ Top ∧ 𝑧 βŠ† π‘₯) β†’ ((π‘₯ ∩ 𝑦) = βˆ… β†’ (𝑧 ∩ 𝑦) = βˆ…))
1110reximdv 3164 . . . 4 ((𝐽 ∈ Top ∧ 𝑧 βŠ† π‘₯) β†’ (βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π·)(π‘₯ ∩ 𝑦) = βˆ… β†’ βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π·)(𝑧 ∩ 𝑦) = βˆ…))
12 simpr 484 . . . . . . 7 ((𝐽 ∈ Top ∧ π‘₯ = 𝑧) β†’ π‘₯ = 𝑧)
1312ineq1d 4206 . . . . . 6 ((𝐽 ∈ Top ∧ π‘₯ = 𝑧) β†’ (π‘₯ ∩ 𝑦) = (𝑧 ∩ 𝑦))
1413eqeq1d 2728 . . . . 5 ((𝐽 ∈ Top ∧ π‘₯ = 𝑧) β†’ ((π‘₯ ∩ 𝑦) = βˆ… ↔ (𝑧 ∩ 𝑦) = βˆ…))
1514rexbidv 3172 . . . 4 ((𝐽 ∈ Top ∧ π‘₯ = 𝑧) β†’ (βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π·)(π‘₯ ∩ 𝑦) = βˆ… ↔ βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π·)(𝑧 ∩ 𝑦) = βˆ…))
162, 11, 15opnneieqv 47817 . . 3 (𝐽 ∈ Top β†’ (βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜πΆ)βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π·)(π‘₯ ∩ 𝑦) = βˆ… ↔ βˆƒπ‘₯ ∈ 𝐽 (𝐢 βŠ† π‘₯ ∧ βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π·)(π‘₯ ∩ 𝑦) = βˆ…)))
17 sepnsepolem1 47828 . . . 4 (βˆƒπ‘₯ ∈ 𝐽 βˆƒπ‘¦ ∈ 𝐽 (𝐢 βŠ† π‘₯ ∧ 𝐷 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…) ↔ βˆƒπ‘₯ ∈ 𝐽 (𝐢 βŠ† π‘₯ ∧ βˆƒπ‘¦ ∈ 𝐽 (𝐷 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…)))
1817a1i 11 . . 3 (𝐽 ∈ Top β†’ (βˆƒπ‘₯ ∈ 𝐽 βˆƒπ‘¦ ∈ 𝐽 (𝐢 βŠ† π‘₯ ∧ 𝐷 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…) ↔ βˆƒπ‘₯ ∈ 𝐽 (𝐢 βŠ† π‘₯ ∧ βˆƒπ‘¦ ∈ 𝐽 (𝐷 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…))))
195, 16, 183bitr4d 311 . 2 (𝐽 ∈ Top β†’ (βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜πΆ)βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π·)(π‘₯ ∩ 𝑦) = βˆ… ↔ βˆƒπ‘₯ ∈ 𝐽 βˆƒπ‘¦ ∈ 𝐽 (𝐢 βŠ† π‘₯ ∧ 𝐷 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…)))
201, 19syl 17 1 (πœ‘ β†’ (βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜πΆ)βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π·)(π‘₯ ∩ 𝑦) = βˆ… ↔ βˆƒπ‘₯ ∈ 𝐽 βˆƒπ‘¦ ∈ 𝐽 (𝐢 βŠ† π‘₯ ∧ 𝐷 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3064   ∩ cin 3942   βŠ† wss 3943  βˆ…c0 4317  β€˜cfv 6537  Topctop 22750  neicnei 22956
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 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-top 22751  df-nei 22957
This theorem is referenced by:  sepcsepo  47833  isnrm4  47837  iscnrm4  47861
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