Theorem uncld 23159

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 4245	. . 3 ⊢ (∪ 𝐽 ∖ (𝐴 ∪ 𝐵)) = ((∪ 𝐽 ∖ 𝐴) ∩ (∪ 𝐽 ∖ 𝐵))
2		cldrcl 23144	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
3		eqid 2765	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	cldopn 23149	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝐴) ∈ 𝐽)
5	3	cldopn 23149	. . . 4 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
6		inopn 23017	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝐴) ∈ 𝐽 ∧ (∪ 𝐽 ∖ 𝐵) ∈ 𝐽) → ((∪ 𝐽 ∖ 𝐴) ∩ (∪ 𝐽 ∖ 𝐵)) ∈ 𝐽)
7	2, 4, 5, 6	syl2an3an 1445	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((∪ 𝐽 ∖ 𝐴) ∩ (∪ 𝐽 ∖ 𝐵)) ∈ 𝐽)
8	1, 7	eqeltrid 2869	. 2 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ (𝐴 ∪ 𝐵)) ∈ 𝐽)
9	3	cldss 23147	. . . . 5 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐴 ⊆ ∪ 𝐽)
10	3	cldss 23147	. . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐵 ⊆ ∪ 𝐽)
11	9, 10	anim12i 624	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ⊆ ∪ 𝐽 ∧ 𝐵 ⊆ ∪ 𝐽))
12		unss 4145	. . . 4 ⊢ ((𝐴 ⊆ ∪ 𝐽 ∧ 𝐵 ⊆ ∪ 𝐽) ↔ (𝐴 ∪ 𝐵) ⊆ ∪ 𝐽)
13	11, 12	sylib 221	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∪ 𝐵) ⊆ ∪ 𝐽)
14	3	iscld2 23146	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝐴 ∪ 𝐵) ⊆ ∪ 𝐽) → ((𝐴 ∪ 𝐵) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (𝐴 ∪ 𝐵)) ∈ 𝐽))
15	2, 13, 14	syl2an2r 697	. 2 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((𝐴 ∪ 𝐵) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (𝐴 ∪ 𝐵)) ∈ 𝐽))
16	8, 15	mpbird 260	1 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∪ 𝐵) ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2145   ∖ cdif 3904   ∪ cun 3905   ∩ cin 3906   ⊆ wss 3907  ∪ cuni 4868  ‘cfv 6525  Topctop 23011  Clsdccld 23134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fn 6528  df-fv 6533  df-top 23012  df-cld 23137

This theorem is referenced by:  iscldtop  23213  paste  23412  lpcls  23482  dvasin  38215  dvacos  38216  dvreasin  38217  dvreacos  38218