Theorem nrmsep 23244

Description: In a normal space, disjoint closed sets are separated by open sets. (Contributed by Jeff Hankins, 1-Feb-2010.)

Assertion

Ref	Expression
nrmsep	⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))

Distinct variable groups:   𝑥,𝑦,𝐶   𝑥,𝐷,𝑦   𝑥,𝐽,𝑦

Proof of Theorem nrmsep

Step	Hyp	Ref	Expression
1		nrmtop 23223	. . . . . 6 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2	1	ad2antrr 726	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐽 ∈ Top)
3		elssuni 4901	. . . . . 6 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
4	3	ad2antrl 728	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝑥 ⊆ ∪ 𝐽)
5		eqid 2729	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
6	5	clscld 22934	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
7	2, 4, 6	syl2anc 584	. . . 4 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
8	5	cldopn 22918	. . . 4 ⊢ (((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∈ 𝐽)
9	7, 8	syl 17	. . 3 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∈ 𝐽)
10		simprrl 780	. . 3 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐶 ⊆ 𝑥)
11		incom 4172	. . . . 5 ⊢ (𝐷 ∩ ((cls‘𝐽)‘𝑥)) = (((cls‘𝐽)‘𝑥) ∩ 𝐷)
12		simprrr 781	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅)
13	11, 12	eqtrid 2776	. . . 4 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (𝐷 ∩ ((cls‘𝐽)‘𝑥)) = ∅)
14		simplr2 1217	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐷 ∈ (Clsd‘𝐽))
15	5	cldss 22916	. . . . 5 ⊢ (𝐷 ∈ (Clsd‘𝐽) → 𝐷 ⊆ ∪ 𝐽)
16		reldisj 4416	. . . . 5 ⊢ (𝐷 ⊆ ∪ 𝐽 → ((𝐷 ∩ ((cls‘𝐽)‘𝑥)) = ∅ ↔ 𝐷 ⊆ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
17	14, 15, 16	3syl 18	. . . 4 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → ((𝐷 ∩ ((cls‘𝐽)‘𝑥)) = ∅ ↔ 𝐷 ⊆ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
18	13, 17	mpbid 232	. . 3 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐷 ⊆ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)))
19	5	sscls 22943	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ ∪ 𝐽) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
20	2, 4, 19	syl2anc 584	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
21	20	ssrind 4207	. . . 4 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) ⊆ (((cls‘𝐽)‘𝑥) ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
22		disjdif 4435	. . . 4 ⊢ (((cls‘𝐽)‘𝑥) ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅
23		sseq0 4366	. . . 4 ⊢ (((𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) ⊆ (((cls‘𝐽)‘𝑥) ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) ∧ (((cls‘𝐽)‘𝑥) ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅) → (𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)
24	21, 22, 23	sylancl 586	. . 3 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)
25		sseq2 3973	. . . . 5 ⊢ (𝑦 = (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → (𝐷 ⊆ 𝑦 ↔ 𝐷 ⊆ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
26		ineq2 4177	. . . . . 6 ⊢ (𝑦 = (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → (𝑥 ∩ 𝑦) = (𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
27	26	eqeq1d 2731	. . . . 5 ⊢ (𝑦 = (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → ((𝑥 ∩ 𝑦) = ∅ ↔ (𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅))
28	25, 27	3anbi23d 1441	. . . 4 ⊢ (𝑦 = (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → ((𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅) ↔ (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∧ (𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)))
29	28	rspcev 3588	. . 3 ⊢ (((∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∧ (𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)) → ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))
30	9, 10, 18, 24, 29	syl13anc 1374	. 2 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))
31		nrmsep2 23243	. 2 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))
32	30, 31	reximddv 3149	1 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ∖ cdif 3911   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  ∪ cuni 4871  ‘cfv 6511  Topctop 22780  Clsdccld 22903  clsccl 22905  Nrmcnrm 23197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-top 22781  df-cld 22906  df-cls 22908  df-nrm 23204

This theorem is referenced by:  isnrm3  23246