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Theorem isnrm4 46224
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 ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥 ∈ ((nei‘𝐽)‘𝑐)∃𝑦 ∈ ((nei‘𝐽)‘𝑑)(𝑥𝑦) = ∅)))
Distinct variable group:   𝐽,𝑐,𝑑,𝑥,𝑦

Proof of Theorem isnrm4
StepHypRef Expression
1 isnrm3 22510 . 2 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))))
2 id 22 . . . . . 6 (𝐽 ∈ Top → 𝐽 ∈ Top)
32sepnsepo 46217 . . . . 5 (𝐽 ∈ Top → (∃𝑥 ∈ ((nei‘𝐽)‘𝑐)∃𝑦 ∈ ((nei‘𝐽)‘𝑑)(𝑥𝑦) = ∅ ↔ ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)))
43imbi2d 341 . . . 4 (𝐽 ∈ Top → (((𝑐𝑑) = ∅ → ∃𝑥 ∈ ((nei‘𝐽)‘𝑐)∃𝑦 ∈ ((nei‘𝐽)‘𝑑)(𝑥𝑦) = ∅) ↔ ((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))))
542ralbidv 3129 . . 3 (𝐽 ∈ Top → (∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥 ∈ ((nei‘𝐽)‘𝑐)∃𝑦 ∈ ((nei‘𝐽)‘𝑑)(𝑥𝑦) = ∅) ↔ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))))
65pm5.32i 575 . 2 ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥 ∈ ((nei‘𝐽)‘𝑐)∃𝑦 ∈ ((nei‘𝐽)‘𝑑)(𝑥𝑦) = ∅)) ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))))
71, 6bitr4i 277 1 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥 ∈ ((nei‘𝐽)‘𝑐)∃𝑦 ∈ ((nei‘𝐽)‘𝑑)(𝑥𝑦) = ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  wrex 3065  cin 3886  wss 3887  c0 4256  cfv 6433  Topctop 22042  Clsdccld 22167  neicnei 22248  Nrmcnrm 22461
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-top 22043  df-cld 22170  df-cls 22172  df-nei 22249  df-nrm 22468
This theorem is referenced by:  dfnrm3  46226
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