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Theorem isnrm3 21443
Description: A topological space is normal iff any two disjoint closed sets are separated by open sets. (Contributed by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
isnrm3 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))))
Distinct variable groups:   𝑥,𝑦   𝑐,𝑑,𝑥,𝑦,𝐽

Proof of Theorem isnrm3
StepHypRef Expression
1 nrmtop 21420 . . 3 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2 nrmsep 21441 . . . . . 6 ((𝐽 ∈ Nrm ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽) ∧ (𝑐𝑑) = ∅)) → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))
323exp2 1463 . . . . 5 (𝐽 ∈ Nrm → (𝑐 ∈ (Clsd‘𝐽) → (𝑑 ∈ (Clsd‘𝐽) → ((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)))))
43impd 398 . . . 4 (𝐽 ∈ Nrm → ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽)) → ((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))))
54ralrimivv 3117 . . 3 (𝐽 ∈ Nrm → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)))
61, 5jca 507 . 2 (𝐽 ∈ Nrm → (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))))
7 simpl 474 . . 3 ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))) → 𝐽 ∈ Top)
8 simpr1 1248 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → 𝑐𝑥)
9 simpr2 1250 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → 𝑑𝑦)
10 sslin 3998 . . . . . . . . . . . . 13 (𝑑𝑦 → (((cls‘𝐽)‘𝑥) ∩ 𝑑) ⊆ (((cls‘𝐽)‘𝑥) ∩ 𝑦))
119, 10syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → (((cls‘𝐽)‘𝑥) ∩ 𝑑) ⊆ (((cls‘𝐽)‘𝑥) ∩ 𝑦))
12 simplll 791 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → 𝐽 ∈ Top)
13 simplr 785 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → 𝑦𝐽)
14 eqid 2765 . . . . . . . . . . . . . . 15 𝐽 = 𝐽
1514opncld 21117 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ( 𝐽𝑦) ∈ (Clsd‘𝐽))
1612, 13, 15syl2anc 579 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → ( 𝐽𝑦) ∈ (Clsd‘𝐽))
17 simpr3 1252 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → (𝑥𝑦) = ∅)
18 simpllr 793 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → 𝑥𝐽)
19 elssuni 4625 . . . . . . . . . . . . . . 15 (𝑥𝐽𝑥 𝐽)
20 reldisj 4181 . . . . . . . . . . . . . . 15 (𝑥 𝐽 → ((𝑥𝑦) = ∅ ↔ 𝑥 ⊆ ( 𝐽𝑦)))
2118, 19, 203syl 18 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → ((𝑥𝑦) = ∅ ↔ 𝑥 ⊆ ( 𝐽𝑦)))
2217, 21mpbid 223 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → 𝑥 ⊆ ( 𝐽𝑦))
2314clsss2 21156 . . . . . . . . . . . . . 14 ((( 𝐽𝑦) ∈ (Clsd‘𝐽) ∧ 𝑥 ⊆ ( 𝐽𝑦)) → ((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝑦))
24 ssdifin0 4210 . . . . . . . . . . . . . 14 (((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝑦) → (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅)
2523, 24syl 17 . . . . . . . . . . . . 13 ((( 𝐽𝑦) ∈ (Clsd‘𝐽) ∧ 𝑥 ⊆ ( 𝐽𝑦)) → (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅)
2616, 22, 25syl2anc 579 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅)
27 sseq0 4137 . . . . . . . . . . . 12 (((((cls‘𝐽)‘𝑥) ∩ 𝑑) ⊆ (((cls‘𝐽)‘𝑥) ∩ 𝑦) ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅) → (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)
2811, 26, 27syl2anc 579 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)
298, 28jca 507 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))
3029ex 401 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) → ((𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅) → (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)))
3130rexlimdva 3178 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑥𝐽) → (∃𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅) → (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)))
3231reximdva 3163 . . . . . . 7 (𝐽 ∈ Top → (∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅) → ∃𝑥𝐽 (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)))
3332imim2d 57 . . . . . 6 (𝐽 ∈ Top → (((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → ((𝑐𝑑) = ∅ → ∃𝑥𝐽 (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))))
3433ralimdv 3110 . . . . 5 (𝐽 ∈ Top → (∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → ∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽 (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))))
3534ralimdv 3110 . . . 4 (𝐽 ∈ Top → (∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽 (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))))
3635imp 395 . . 3 ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))) → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽 (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)))
37 isnrm2 21442 . . 3 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽 (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))))
387, 36, 37sylanbrc 578 . 2 ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))) → 𝐽 ∈ Nrm)
396, 38impbii 200 1 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  wrex 3056  cdif 3729  cin 3731  wss 3732  c0 4079   cuni 4594  cfv 6068  Topctop 20977  Clsdccld 21100  clsccl 21102  Nrmcnrm 21394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-top 20978  df-cld 21103  df-cls 21105  df-nrm 21401
This theorem is referenced by:  metnrm  22944
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