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Theorem isnrm3 22582
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 22559 . . 3 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2 nrmsep 22580 . . . . . 6 ((𝐽 ∈ Nrm ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽) ∧ (𝑐𝑑) = ∅)) → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))
323exp2 1353 . . . . 5 (𝐽 ∈ Nrm → (𝑐 ∈ (Clsd‘𝐽) → (𝑑 ∈ (Clsd‘𝐽) → ((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)))))
43impd 411 . . . 4 (𝐽 ∈ Nrm → ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽)) → ((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))))
54ralrimivv 3192 . . 3 (𝐽 ∈ Nrm → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)))
61, 5jca 512 . 2 (𝐽 ∈ Nrm → (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))))
7 simpl 483 . . 3 ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))) → 𝐽 ∈ Top)
8 simpr1 1193 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → 𝑐𝑥)
9 simpr2 1194 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → 𝑑𝑦)
10 sslin 4179 . . . . . . . . . . . 12 (𝑑𝑦 → (((cls‘𝐽)‘𝑥) ∩ 𝑑) ⊆ (((cls‘𝐽)‘𝑥) ∩ 𝑦))
119, 10syl 17 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → (((cls‘𝐽)‘𝑥) ∩ 𝑑) ⊆ (((cls‘𝐽)‘𝑥) ∩ 𝑦))
12 eqid 2737 . . . . . . . . . . . . . 14 𝐽 = 𝐽
1312opncld 22256 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ( 𝐽𝑦) ∈ (Clsd‘𝐽))
1413ad4ant13 748 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → ( 𝐽𝑦) ∈ (Clsd‘𝐽))
15 simpr3 1195 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → (𝑥𝑦) = ∅)
16 simpllr 773 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → 𝑥𝐽)
17 elssuni 4883 . . . . . . . . . . . . . 14 (𝑥𝐽𝑥 𝐽)
18 reldisj 4396 . . . . . . . . . . . . . 14 (𝑥 𝐽 → ((𝑥𝑦) = ∅ ↔ 𝑥 ⊆ ( 𝐽𝑦)))
1916, 17, 183syl 18 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → ((𝑥𝑦) = ∅ ↔ 𝑥 ⊆ ( 𝐽𝑦)))
2015, 19mpbid 231 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → 𝑥 ⊆ ( 𝐽𝑦))
2112clsss2 22295 . . . . . . . . . . . . 13 ((( 𝐽𝑦) ∈ (Clsd‘𝐽) ∧ 𝑥 ⊆ ( 𝐽𝑦)) → ((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝑦))
22 ssdifin0 4428 . . . . . . . . . . . . 13 (((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝑦) → (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅)
2321, 22syl 17 . . . . . . . . . . . 12 ((( 𝐽𝑦) ∈ (Clsd‘𝐽) ∧ 𝑥 ⊆ ( 𝐽𝑦)) → (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅)
2414, 20, 23syl2anc 584 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅)
25 sseq0 4344 . . . . . . . . . . 11 (((((cls‘𝐽)‘𝑥) ∩ 𝑑) ⊆ (((cls‘𝐽)‘𝑥) ∩ 𝑦) ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅) → (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)
2611, 24, 25syl2anc 584 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)
278, 26jca 512 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))
2827rexlimdva2 3151 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑥𝐽) → (∃𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅) → (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)))
2928reximdva 3162 . . . . . . 7 (𝐽 ∈ Top → (∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅) → ∃𝑥𝐽 (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)))
3029imim2d 57 . . . . . 6 (𝐽 ∈ Top → (((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → ((𝑐𝑑) = ∅ → ∃𝑥𝐽 (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))))
3130ralimdv 3163 . . . . 5 (𝐽 ∈ Top → (∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → ∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽 (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))))
3231ralimdv 3163 . . . 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) = ∅))))
3332imp 407 . . 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) = ∅)))
34 isnrm2 22581 . . 3 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽 (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))))
357, 33, 34sylanbrc 583 . 2 ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))) → 𝐽 ∈ Nrm)
366, 35impbii 208 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 205  wa 396  w3a 1086   = wceq 1540  wcel 2105  wral 3062  wrex 3071  cdif 3894  cin 3896  wss 3897  c0 4267   cuni 4850  cfv 6465  Topctop 22114  Clsdccld 22239  clsccl 22241  Nrmcnrm 22533
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 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7628
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-int 4893  df-iun 4939  df-iin 4940  df-br 5088  df-opab 5150  df-mpt 5171  df-id 5507  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-top 22115  df-cld 22242  df-cls 22244  df-nrm 22540
This theorem is referenced by:  metnrm  24097  isnrm4  46476  dfnrm2  46477  iscnrm3  46498
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