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Theorem isnrm4 47727
Description: A topological space is normal iff any two disjoint closed sets are separated by neighborhoods. (Contributed by Zhi Wang, 1-Sep-2024.)
Assertion
Ref Expression
isnrm4 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ βˆ€π‘ ∈ (Clsdβ€˜π½)βˆ€π‘‘ ∈ (Clsdβ€˜π½)((𝑐 ∩ 𝑑) = βˆ… β†’ βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜π‘)βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π‘‘)(π‘₯ ∩ 𝑦) = βˆ…)))
Distinct variable group:   𝐽,𝑐,𝑑,π‘₯,𝑦

Proof of Theorem isnrm4
StepHypRef Expression
1 isnrm3 23183 . 2 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ βˆ€π‘ ∈ (Clsdβ€˜π½)βˆ€π‘‘ ∈ (Clsdβ€˜π½)((𝑐 ∩ 𝑑) = βˆ… β†’ βˆƒπ‘₯ ∈ 𝐽 βˆƒπ‘¦ ∈ 𝐽 (𝑐 βŠ† π‘₯ ∧ 𝑑 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…))))
2 id 22 . . . . . 6 (𝐽 ∈ Top β†’ 𝐽 ∈ Top)
32sepnsepo 47720 . . . . 5 (𝐽 ∈ Top β†’ (βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜π‘)βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π‘‘)(π‘₯ ∩ 𝑦) = βˆ… ↔ βˆƒπ‘₯ ∈ 𝐽 βˆƒπ‘¦ ∈ 𝐽 (𝑐 βŠ† π‘₯ ∧ 𝑑 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…)))
43imbi2d 340 . . . 4 (𝐽 ∈ Top β†’ (((𝑐 ∩ 𝑑) = βˆ… β†’ βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜π‘)βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π‘‘)(π‘₯ ∩ 𝑦) = βˆ…) ↔ ((𝑐 ∩ 𝑑) = βˆ… β†’ βˆƒπ‘₯ ∈ 𝐽 βˆƒπ‘¦ ∈ 𝐽 (𝑐 βŠ† π‘₯ ∧ 𝑑 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…))))
542ralbidv 3217 . . 3 (𝐽 ∈ Top β†’ (βˆ€π‘ ∈ (Clsdβ€˜π½)βˆ€π‘‘ ∈ (Clsdβ€˜π½)((𝑐 ∩ 𝑑) = βˆ… β†’ βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜π‘)βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π‘‘)(π‘₯ ∩ 𝑦) = βˆ…) ↔ βˆ€π‘ ∈ (Clsdβ€˜π½)βˆ€π‘‘ ∈ (Clsdβ€˜π½)((𝑐 ∩ 𝑑) = βˆ… β†’ βˆƒπ‘₯ ∈ 𝐽 βˆƒπ‘¦ ∈ 𝐽 (𝑐 βŠ† π‘₯ ∧ 𝑑 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…))))
65pm5.32i 574 . 2 ((𝐽 ∈ Top ∧ βˆ€π‘ ∈ (Clsdβ€˜π½)βˆ€π‘‘ ∈ (Clsdβ€˜π½)((𝑐 ∩ 𝑑) = βˆ… β†’ βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜π‘)βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π‘‘)(π‘₯ ∩ 𝑦) = βˆ…)) ↔ (𝐽 ∈ Top ∧ βˆ€π‘ ∈ (Clsdβ€˜π½)βˆ€π‘‘ ∈ (Clsdβ€˜π½)((𝑐 ∩ 𝑑) = βˆ… β†’ βˆƒπ‘₯ ∈ 𝐽 βˆƒπ‘¦ ∈ 𝐽 (𝑐 βŠ† π‘₯ ∧ 𝑑 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…))))
71, 6bitr4i 278 1 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ βˆ€π‘ ∈ (Clsdβ€˜π½)βˆ€π‘‘ ∈ (Clsdβ€˜π½)((𝑐 ∩ 𝑑) = βˆ… β†’ βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜π‘)βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π‘‘)(π‘₯ ∩ 𝑦) = βˆ…)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  βˆƒwrex 3069   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  β€˜cfv 6543  Topctop 22715  Clsdccld 22840  neicnei 22921  Nrmcnrm 23134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 22716  df-cld 22843  df-cls 22845  df-nei 22922  df-nrm 23141
This theorem is referenced by:  dfnrm3  47729
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