Theorem uncld 14292

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 3412	. . 3 ⊢ (∪ 𝐽 ∖ (𝐴 ∪ 𝐵)) = ((∪ 𝐽 ∖ 𝐴) ∩ (∪ 𝐽 ∖ 𝐵))
2		cldrcl 14281	. . . . 5 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
3	2	adantr 276	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
4		eqid 2193	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
5	4	cldopn 14286	. . . . 5 ⊢ (𝐴 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝐴) ∈ 𝐽)
6	5	adantr 276	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝐴) ∈ 𝐽)
7	4	cldopn 14286	. . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
8	7	adantl 277	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
9		inopn 14182	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝐴) ∈ 𝐽 ∧ (∪ 𝐽 ∖ 𝐵) ∈ 𝐽) → ((∪ 𝐽 ∖ 𝐴) ∩ (∪ 𝐽 ∖ 𝐵)) ∈ 𝐽)
10	3, 6, 8, 9	syl3anc 1249	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((∪ 𝐽 ∖ 𝐴) ∩ (∪ 𝐽 ∖ 𝐵)) ∈ 𝐽)
11	1, 10	eqeltrid 2280	. 2 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ (𝐴 ∪ 𝐵)) ∈ 𝐽)
12	4	cldss 14284	. . . . 5 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐴 ⊆ ∪ 𝐽)
13	4	cldss 14284	. . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐵 ⊆ ∪ 𝐽)
14	12, 13	anim12i 338	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ⊆ ∪ 𝐽 ∧ 𝐵 ⊆ ∪ 𝐽))
15		unss 3334	. . . 4 ⊢ ((𝐴 ⊆ ∪ 𝐽 ∧ 𝐵 ⊆ ∪ 𝐽) ↔ (𝐴 ∪ 𝐵) ⊆ ∪ 𝐽)
16	14, 15	sylib 122	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∪ 𝐵) ⊆ ∪ 𝐽)
17	4	iscld2 14283	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝐴 ∪ 𝐵) ⊆ ∪ 𝐽) → ((𝐴 ∪ 𝐵) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (𝐴 ∪ 𝐵)) ∈ 𝐽))
18	3, 16, 17	syl2anc 411	. 2 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((𝐴 ∪ 𝐵) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (𝐴 ∪ 𝐵)) ∈ 𝐽))
19	11, 18	mpbird 167	1 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∪ 𝐵) ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2164   ∖ cdif 3151   ∪ cun 3152   ∩ cin 3153   ⊆ wss 3154  ∪ cuni 3836  ‘cfv 5255  Topctop 14176  Clsdccld 14271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-top 14177  df-cld 14274

This theorem is referenced by:  iuncld  14294