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Theorem isnrm3 21968
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 21945 . . 3 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2 nrmsep 21966 . . . . . 6 ((𝐽 ∈ Nrm ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽) ∧ (𝑐𝑑) = ∅)) → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))
323exp2 1351 . . . . 5 (𝐽 ∈ Nrm → (𝑐 ∈ (Clsd‘𝐽) → (𝑑 ∈ (Clsd‘𝐽) → ((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)))))
43impd 414 . . . 4 (𝐽 ∈ Nrm → ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽)) → ((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))))
54ralrimivv 3158 . . 3 (𝐽 ∈ Nrm → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)))
61, 5jca 515 . 2 (𝐽 ∈ Nrm → (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))))
7 simpl 486 . . 3 ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))) → 𝐽 ∈ Top)
8 simpr1 1191 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → 𝑐𝑥)
9 simpr2 1192 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → 𝑑𝑦)
10 sslin 4164 . . . . . . . . . . . 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 2801 . . . . . . . . . . . . . 14 𝐽 = 𝐽
1312opncld 21642 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ( 𝐽𝑦) ∈ (Clsd‘𝐽))
1413ad4ant13 750 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → ( 𝐽𝑦) ∈ (Clsd‘𝐽))
15 simpr3 1193 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → (𝑥𝑦) = ∅)
16 simpllr 775 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → 𝑥𝐽)
17 elssuni 4833 . . . . . . . . . . . . . 14 (𝑥𝐽𝑥 𝐽)
18 reldisj 4362 . . . . . . . . . . . . . 14 (𝑥 𝐽 → ((𝑥𝑦) = ∅ ↔ 𝑥 ⊆ ( 𝐽𝑦)))
1916, 17, 183syl 18 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → ((𝑥𝑦) = ∅ ↔ 𝑥 ⊆ ( 𝐽𝑦)))
2015, 19mpbid 235 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → 𝑥 ⊆ ( 𝐽𝑦))
2112clsss2 21681 . . . . . . . . . . . . 13 ((( 𝐽𝑦) ∈ (Clsd‘𝐽) ∧ 𝑥 ⊆ ( 𝐽𝑦)) → ((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝑦))
22 ssdifin0 4392 . . . . . . . . . . . . 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 587 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅)
25 sseq0 4310 . . . . . . . . . . 11 (((((cls‘𝐽)‘𝑥) ∩ 𝑑) ⊆ (((cls‘𝐽)‘𝑥) ∩ 𝑦) ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅) → (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)
2611, 24, 25syl2anc 587 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)
278, 26jca 515 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))
2827rexlimdva2 3249 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑥𝐽) → (∃𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅) → (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)))
2928reximdva 3236 . . . . . . 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 3148 . . . . 5 (𝐽 ∈ Top → (∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → ∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽 (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))))
3231ralimdv 3148 . . . 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 410 . . 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 21967 . . 3 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽 (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))))
357, 33, 34sylanbrc 586 . 2 ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))) → 𝐽 ∈ Nrm)
366, 35impbii 212 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  wrex 3110  cdif 3881  cin 3883  wss 3884  c0 4246   cuni 4803  cfv 6328  Topctop 21502  Clsdccld 21625  clsccl 21627  Nrmcnrm 21919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-top 21503  df-cld 21628  df-cls 21630  df-nrm 21926
This theorem is referenced by:  metnrm  23471
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