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Theorem sepnsepo 47042
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 47041 . . . . 5 (𝐽 ∈ Top β†’ (βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π·)(π‘₯ ∩ 𝑦) = βˆ… ↔ βˆƒπ‘¦ ∈ 𝐽 (𝐷 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…)))
43anbi2d 630 . . . 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 4194 . . . . . . 7 (𝑧 βŠ† π‘₯ β†’ (𝑧 ∩ 𝑦) βŠ† (π‘₯ ∩ 𝑦))
7 sseq0 4360 . . . . . . . 8 (((𝑧 ∩ 𝑦) βŠ† (π‘₯ ∩ 𝑦) ∧ (π‘₯ ∩ 𝑦) = βˆ…) β†’ (𝑧 ∩ 𝑦) = βˆ…)
87ex 414 . . . . . . 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 483 . . . . 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 486 . . . . . . 7 ((𝐽 ∈ Top ∧ π‘₯ = 𝑧) β†’ π‘₯ = 𝑧)
1312ineq1d 4172 . . . . . 6 ((𝐽 ∈ Top ∧ π‘₯ = 𝑧) β†’ (π‘₯ ∩ 𝑦) = (𝑧 ∩ 𝑦))
1413eqeq1d 2735 . . . . 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 47029 . . 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 47040 . . . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3070   ∩ cin 3910   βŠ† wss 3911  βˆ…c0 4283  β€˜cfv 6497  Topctop 22258  neicnei 22464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-top 22259  df-nei 22465
This theorem is referenced by:  sepcsepo  47045  isnrm4  47049  iscnrm4  47073
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