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Theorem uncld 21223
 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)) → (𝐴𝐵) ∈ (Clsd‘𝐽))

Proof of Theorem uncld
StepHypRef Expression
1 difundi 4111 . . 3 ( 𝐽 ∖ (𝐴𝐵)) = (( 𝐽𝐴) ∩ ( 𝐽𝐵))
2 cldrcl 21208 . . . . 5 (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
32adantr 474 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
4 eqid 2825 . . . . . 6 𝐽 = 𝐽
54cldopn 21213 . . . . 5 (𝐴 ∈ (Clsd‘𝐽) → ( 𝐽𝐴) ∈ 𝐽)
65adantr 474 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ( 𝐽𝐴) ∈ 𝐽)
74cldopn 21213 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → ( 𝐽𝐵) ∈ 𝐽)
87adantl 475 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ( 𝐽𝐵) ∈ 𝐽)
9 inopn 21081 . . . 4 ((𝐽 ∈ Top ∧ ( 𝐽𝐴) ∈ 𝐽 ∧ ( 𝐽𝐵) ∈ 𝐽) → (( 𝐽𝐴) ∩ ( 𝐽𝐵)) ∈ 𝐽)
103, 6, 8, 9syl3anc 1494 . . 3 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (( 𝐽𝐴) ∩ ( 𝐽𝐵)) ∈ 𝐽)
111, 10syl5eqel 2910 . 2 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ( 𝐽 ∖ (𝐴𝐵)) ∈ 𝐽)
124cldss 21211 . . . . 5 (𝐴 ∈ (Clsd‘𝐽) → 𝐴 𝐽)
134cldss 21211 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → 𝐵 𝐽)
1412, 13anim12i 606 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 𝐽𝐵 𝐽))
15 unss 4016 . . . 4 ((𝐴 𝐽𝐵 𝐽) ↔ (𝐴𝐵) ⊆ 𝐽)
1614, 15sylib 210 . . 3 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝐵) ⊆ 𝐽)
174iscld2 21210 . . 3 ((𝐽 ∈ Top ∧ (𝐴𝐵) ⊆ 𝐽) → ((𝐴𝐵) ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ (𝐴𝐵)) ∈ 𝐽))
183, 16, 17syl2anc 579 . 2 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((𝐴𝐵) ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ (𝐴𝐵)) ∈ 𝐽))
1911, 18mpbird 249 1 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝐵) ∈ (Clsd‘𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∈ wcel 2164   ∖ cdif 3795   ∪ cun 3796   ∩ cin 3797   ⊆ wss 3798  ∪ cuni 4660  ‘cfv 6127  Topctop 21075  Clsdccld 21198 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-iota 6090  df-fun 6129  df-fn 6130  df-fv 6135  df-top 21076  df-cld 21201 This theorem is referenced by:  iscldtop  21277  paste  21476  lpcls  21546  dvasin  34038  dvacos  34039  dvreasin  34040  dvreacos  34041
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