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Theorem isnrm3 23383
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 23360 . . 3 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2 nrmsep 23381 . . . . . 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 410 . . . 4 (𝐽 ∈ Nrm → ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽)) → ((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))))
54ralrimivv 3198 . . 3 (𝐽 ∈ Nrm → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)))
61, 5jca 511 . 2 (𝐽 ∈ Nrm → (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))))
7 simpl 482 . . 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 4251 . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . 14 𝐽 = 𝐽
1312opncld 23057 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ( 𝐽𝑦) ∈ (Clsd‘𝐽))
1413ad4ant13 751 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → ( 𝐽𝑦) ∈ (Clsd‘𝐽))
15 simpr3 1195 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → (𝑥𝑦) = ∅)
16 simpllr 776 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → 𝑥𝐽)
17 elssuni 4942 . . . . . . . . . . . . . 14 (𝑥𝐽𝑥 𝐽)
18 reldisj 4459 . . . . . . . . . . . . . 14 (𝑥 𝐽 → ((𝑥𝑦) = ∅ ↔ 𝑥 ⊆ ( 𝐽𝑦)))
1916, 17, 183syl 18 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → ((𝑥𝑦) = ∅ ↔ 𝑥 ⊆ ( 𝐽𝑦)))
2015, 19mpbid 232 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → 𝑥 ⊆ ( 𝐽𝑦))
2112clsss2 23096 . . . . . . . . . . . . 13 ((( 𝐽𝑦) ∈ (Clsd‘𝐽) ∧ 𝑥 ⊆ ( 𝐽𝑦)) → ((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝑦))
22 ssdifin0 4492 . . . . . . . . . . . . 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 4409 . . . . . . . . . . 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 511 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))
2827rexlimdva2 3155 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑥𝐽) → (∃𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅) → (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)))
2928reximdva 3166 . . . . . . 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 3167 . . . . 5 (𝐽 ∈ Top → (∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → ∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽 (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))))
3231ralimdv 3167 . . . 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 406 . . 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 23382 . . 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 209 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  cdif 3960  cin 3962  wss 3963  c0 4339   cuni 4912  cfv 6563  Topctop 22915  Clsdccld 23040  clsccl 23042  Nrmcnrm 23334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-top 22916  df-cld 23043  df-cls 23045  df-nrm 23341
This theorem is referenced by:  metnrm  24898  isnrm4  48727  dfnrm2  48728  iscnrm3  48749
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