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Theorem isnrm3 21365
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 21342 . . 3 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2 nrmsep 21363 . . . . . 6 ((𝐽 ∈ Nrm ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽) ∧ (𝑐𝑑) = ∅)) → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))
323exp2 1448 . . . . 5 (𝐽 ∈ Nrm → (𝑐 ∈ (Clsd‘𝐽) → (𝑑 ∈ (Clsd‘𝐽) → ((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)))))
43impd 446 . . . 4 (𝐽 ∈ Nrm → ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽)) → ((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))))
54ralrimivv 3108 . . 3 (𝐽 ∈ Nrm → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)))
61, 5jca 555 . 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 1234 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → 𝑐𝑥)
9 simpr2 1236 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → 𝑑𝑦)
10 sslin 3982 . . . . . . . . . . . . 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 815 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → 𝐽 ∈ Top)
13 simplr 809 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → 𝑦𝐽)
14 eqid 2760 . . . . . . . . . . . . . . 15 𝐽 = 𝐽
1514opncld 21039 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ( 𝐽𝑦) ∈ (Clsd‘𝐽))
1612, 13, 15syl2anc 696 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → ( 𝐽𝑦) ∈ (Clsd‘𝐽))
17 simpr3 1238 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → (𝑥𝑦) = ∅)
18 simpllr 817 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → 𝑥𝐽)
19 elssuni 4619 . . . . . . . . . . . . . . 15 (𝑥𝐽𝑥 𝐽)
20 reldisj 4163 . . . . . . . . . . . . . . 15 (𝑥 𝐽 → ((𝑥𝑦) = ∅ ↔ 𝑥 ⊆ ( 𝐽𝑦)))
2118, 19, 203syl 18 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → ((𝑥𝑦) = ∅ ↔ 𝑥 ⊆ ( 𝐽𝑦)))
2217, 21mpbid 222 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → 𝑥 ⊆ ( 𝐽𝑦))
2314clsss2 21078 . . . . . . . . . . . . . 14 ((( 𝐽𝑦) ∈ (Clsd‘𝐽) ∧ 𝑥 ⊆ ( 𝐽𝑦)) → ((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝑦))
24 ssdifin0 4194 . . . . . . . . . . . . . 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 696 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅)
27 sseq0 4118 . . . . . . . . . . . 12 (((((cls‘𝐽)‘𝑥) ∩ 𝑑) ⊆ (((cls‘𝐽)‘𝑥) ∩ 𝑦) ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅) → (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)
2811, 26, 27syl2anc 696 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)
298, 28jca 555 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) ∧ (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))
3029ex 449 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑦𝐽) → ((𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅) → (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)))
3130rexlimdva 3169 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑥𝐽) → (∃𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅) → (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)))
3231reximdva 3155 . . . . . . 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 3101 . . . . 5 (𝐽 ∈ Top → (∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅)) → ∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽 (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))))
3534ralimdv 3101 . . . 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 444 . . 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 21364 . . 3 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽 (𝑐𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))))
387, 36, 37sylanbrc 701 . 2 ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))) → 𝐽 ∈ Nrm)
396, 38impbii 199 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wrex 3051  cdif 3712  cin 3714  wss 3715  c0 4058   cuni 4588  cfv 6049  Topctop 20900  Clsdccld 21022  clsccl 21024  Nrmcnrm 21316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-top 20901  df-cld 21025  df-cls 21027  df-nrm 21323
This theorem is referenced by:  metnrm  22866
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