Theorem nrmsep 11615

Description: In a normal space, disjoint closed sets are separated by open sets.

Assertion

Ref	Expression
nrmsep	⊢ ((J ∈ Nrm ⋀ (C ∈ (Clsd ‘J) ⋀ D ∈ (Clsd ‘J) ⋀ (C ∩ D) = ∅)) → ∃o ∈ J ∃p ∈ J (C ⊆ o ⋀ D ⊆ p ⋀ (o ∩ p) = ∅))

Distinct variable groups:   o,p,C   D,o,p   o,J,p

Proof of Theorem nrmsep

Step	Hyp	Ref	Expression
1		isnrm 11612	. . . 4 ⊢ (J ∈ Nrm ↔ (J ∈ Top ⋀ ∀c ∈ (Clsd ‘J)∀d ∈ (Clsd ‘J)((c ∩ d) = ∅ → ∃o ∈ J ∃p ∈ J (c ⊆ o ⋀ d ⊆ p ⋀ (o ∩ p) = ∅))))
2	1	pm3.27bi 324	. . 3 ⊢ (J ∈ Nrm → ∀c ∈ (Clsd ‘J)∀d ∈ (Clsd ‘J)((c ∩ d) = ∅ → ∃o ∈ J ∃p ∈ J (c ⊆ o ⋀ d ⊆ p ⋀ (o ∩ p) = ∅)))
3		ineq1 2262	. . . . . . . . 9 ⊢ (c = C → (c ∩ d) = (C ∩ d))
4	3	eqeq1d 1526	. . . . . . . 8 ⊢ (c = C → ((c ∩ d) = ∅ ↔ (C ∩ d) = ∅))
5		sseq1 2134	. . . . . . . . . 10 ⊢ (c = C → (c ⊆ o ↔ C ⊆ o))
6	5	3anbi1d 903	. . . . . . . . 9 ⊢ (c = C → ((c ⊆ o ⋀ d ⊆ p ⋀ (o ∩ p) = ∅) ↔ (C ⊆ o ⋀ d ⊆ p ⋀ (o ∩ p) = ∅)))
7	6	2rexbidv 1727	. . . . . . . 8 ⊢ (c = C → (∃o ∈ J ∃p ∈ J (c ⊆ o ⋀ d ⊆ p ⋀ (o ∩ p) = ∅) ↔ ∃o ∈ J ∃p ∈ J (C ⊆ o ⋀ d ⊆ p ⋀ (o ∩ p) = ∅)))
8	4, 7	imbi12d 629	. . . . . . 7 ⊢ (c = C → (((c ∩ d) = ∅ → ∃o ∈ J ∃p ∈ J (c ⊆ o ⋀ d ⊆ p ⋀ (o ∩ p) = ∅)) ↔ ((C ∩ d) = ∅ → ∃o ∈ J ∃p ∈ J (C ⊆ o ⋀ d ⊆ p ⋀ (o ∩ p) = ∅))))
9		ineq2 2263	. . . . . . . . 9 ⊢ (d = D → (C ∩ d) = (C ∩ D))
10	9	eqeq1d 1526	. . . . . . . 8 ⊢ (d = D → ((C ∩ d) = ∅ ↔ (C ∩ D) = ∅))
11		sseq1 2134	. . . . . . . . . 10 ⊢ (d = D → (d ⊆ p ↔ D ⊆ p))
12	11	3anbi2d 904	. . . . . . . . 9 ⊢ (d = D → ((C ⊆ o ⋀ d ⊆ p ⋀ (o ∩ p) = ∅) ↔ (C ⊆ o ⋀ D ⊆ p ⋀ (o ∩ p) = ∅)))
13	12	2rexbidv 1727	. . . . . . . 8 ⊢ (d = D → (∃o ∈ J ∃p ∈ J (C ⊆ o ⋀ d ⊆ p ⋀ (o ∩ p) = ∅) ↔ ∃o ∈ J ∃p ∈ J (C ⊆ o ⋀ D ⊆ p ⋀ (o ∩ p) = ∅)))
14	10, 13	imbi12d 629	. . . . . . 7 ⊢ (d = D → (((C ∩ d) = ∅ → ∃o ∈ J ∃p ∈ J (C ⊆ o ⋀ d ⊆ p ⋀ (o ∩ p) = ∅)) ↔ ((C ∩ D) = ∅ → ∃o ∈ J ∃p ∈ J (C ⊆ o ⋀ D ⊆ p ⋀ (o ∩ p) = ∅))))
15	8, 14	rcla42v 1926	. . . . . 6 ⊢ ((C ∈ (Clsd ‘J) ⋀ D ∈ (Clsd ‘J)) → (∀c ∈ (Clsd ‘J)∀d ∈ (Clsd ‘J)((c ∩ d) = ∅ → ∃o ∈ J ∃p ∈ J (c ⊆ o ⋀ d ⊆ p ⋀ (o ∩ p) = ∅)) → ((C ∩ D) = ∅ → ∃o ∈ J ∃p ∈ J (C ⊆ o ⋀ D ⊆ p ⋀ (o ∩ p) = ∅))))
16	15	com12 11	. . . . 5 ⊢ (∀c ∈ (Clsd ‘J)∀d ∈ (Clsd ‘J)((c ∩ d) = ∅ → ∃o ∈ J ∃p ∈ J (c ⊆ o ⋀ d ⊆ p ⋀ (o ∩ p) = ∅)) → ((C ∈ (Clsd ‘J) ⋀ D ∈ (Clsd ‘J)) → ((C ∩ D) = ∅ → ∃o ∈ J ∃p ∈ J (C ⊆ o ⋀ D ⊆ p ⋀ (o ∩ p) = ∅))))
17	16	exp3a 374	. . . 4 ⊢ (∀c ∈ (Clsd ‘J)∀d ∈ (Clsd ‘J)((c ∩ d) = ∅ → ∃o ∈ J ∃p ∈ J (c ⊆ o ⋀ d ⊆ p ⋀ (o ∩ p) = ∅)) → (C ∈ (Clsd ‘J) → (D ∈ (Clsd ‘J) → ((C ∩ D) = ∅ → ∃o ∈ J ∃p ∈ J (C ⊆ o ⋀ D ⊆ p ⋀ (o ∩ p) = ∅)))))
18	17	3impd 853	. . 3 ⊢ (∀c ∈ (Clsd ‘J)∀d ∈ (Clsd ‘J)((c ∩ d) = ∅ → ∃o ∈ J ∃p ∈ J (c ⊆ o ⋀ d ⊆ p ⋀ (o ∩ p) = ∅)) → ((C ∈ (Clsd ‘J) ⋀ D ∈ (Clsd ‘J) ⋀ (C ∩ D) = ∅) → ∃o ∈ J ∃p ∈ J (C ⊆ o ⋀ D ⊆ p ⋀ (o ∩ p) = ∅)))
19	2, 18	syl 10	. 2 ⊢ (J ∈ Nrm → ((C ∈ (Clsd ‘J) ⋀ D ∈ (Clsd ‘J) ⋀ (C ∩ D) = ∅) → ∃o ∈ J ∃p ∈ J (C ⊆ o ⋀ D ⊆ p ⋀ (o ∩ p) = ∅)))
20	19	imp 348	1 ⊢ ((J ∈ Nrm ⋀ (C ∈ (Clsd ‘J) ⋀ D ∈ (Clsd ‘J) ⋀ (C ∩ D) = ∅)) → ∃o ∈ J ∃p ∈ J (C ⊆ o ⋀ D ⊆ p ⋀ (o ∩ p) = ∅))

Syntax hints:   → wi 3   ⋀ wa 221   ⋀ w3a 781   = wceq 992   ∈ wcel 994  ∀wral 1691  ∃wrex 1692   ∩ cin 2098   ⊆ wss 2099  ∅c0 2332   ‘cfv 3263  Topctop 7800  Clsdccld 7870  Nrmcnrm 11595

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-pow 2818  ax-pr 2855

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-rab 1698  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-xp 3265  df-cnv 3267  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fv 3279  df-nrm 11598

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