Theorem uncld 21067

Description: The union of two closed sets is closed. Equivalent to Theorem 6.1(3) of [Munkres] p. 93. (Contributed by NM, 5-Oct-2006.)

Assertion

Ref	Expression
uncld	⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∪ 𝐵) ∈ (Clsd‘𝐽))

Proof of Theorem uncld

Step	Hyp	Ref	Expression
1		difundi 4022	. . 3 ⊢ (∪ 𝐽 ∖ (𝐴 ∪ 𝐵)) = ((∪ 𝐽 ∖ 𝐴) ∩ (∪ 𝐽 ∖ 𝐵))
2		cldrcl 21052	. . . . 5 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
3	2	adantr 472	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
4		eqid 2760	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
5	4	cldopn 21057	. . . . 5 ⊢ (𝐴 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝐴) ∈ 𝐽)
6	5	adantr 472	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝐴) ∈ 𝐽)
7	4	cldopn 21057	. . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
8	7	adantl 473	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
9		inopn 20926	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝐴) ∈ 𝐽 ∧ (∪ 𝐽 ∖ 𝐵) ∈ 𝐽) → ((∪ 𝐽 ∖ 𝐴) ∩ (∪ 𝐽 ∖ 𝐵)) ∈ 𝐽)
10	3, 6, 8, 9	syl3anc 1477	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((∪ 𝐽 ∖ 𝐴) ∩ (∪ 𝐽 ∖ 𝐵)) ∈ 𝐽)
11	1, 10	syl5eqel 2843	. 2 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ (𝐴 ∪ 𝐵)) ∈ 𝐽)
12	4	cldss 21055	. . . . 5 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐴 ⊆ ∪ 𝐽)
13	4	cldss 21055	. . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐵 ⊆ ∪ 𝐽)
14	12, 13	anim12i 591	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ⊆ ∪ 𝐽 ∧ 𝐵 ⊆ ∪ 𝐽))
15		unss 3930	. . . 4 ⊢ ((𝐴 ⊆ ∪ 𝐽 ∧ 𝐵 ⊆ ∪ 𝐽) ↔ (𝐴 ∪ 𝐵) ⊆ ∪ 𝐽)
16	14, 15	sylib 208	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∪ 𝐵) ⊆ ∪ 𝐽)
17	4	iscld2 21054	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝐴 ∪ 𝐵) ⊆ ∪ 𝐽) → ((𝐴 ∪ 𝐵) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (𝐴 ∪ 𝐵)) ∈ 𝐽))
18	3, 16, 17	syl2anc 696	. 2 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((𝐴 ∪ 𝐵) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (𝐴 ∪ 𝐵)) ∈ 𝐽))
19	11, 18	mpbird 247	1 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∪ 𝐵) ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∈ wcel 2139   ∖ cdif 3712   ∪ cun 3713   ∩ cin 3714   ⊆ wss 3715  ∪ cuni 4588  ‘cfv 6049  Topctop 20920  Clsdccld 21042

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-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115

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-rab 3059  df-v 3342  df-sbc 3577  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-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-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-top 20921  df-cld 21045

This theorem is referenced by:  iscldtop  21121  paste  21320  lpcls  21390  dvasin  33827  dvacos  33828  dvreasin  33829  dvreacos  33830