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Theorem uncld 12271
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 3323 . . 3 ( 𝐽 ∖ (𝐴𝐵)) = (( 𝐽𝐴) ∩ ( 𝐽𝐵))
2 cldrcl 12260 . . . . 5 (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
32adantr 274 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
4 eqid 2137 . . . . . 6 𝐽 = 𝐽
54cldopn 12265 . . . . 5 (𝐴 ∈ (Clsd‘𝐽) → ( 𝐽𝐴) ∈ 𝐽)
65adantr 274 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ( 𝐽𝐴) ∈ 𝐽)
74cldopn 12265 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → ( 𝐽𝐵) ∈ 𝐽)
87adantl 275 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ( 𝐽𝐵) ∈ 𝐽)
9 inopn 12159 . . . 4 ((𝐽 ∈ Top ∧ ( 𝐽𝐴) ∈ 𝐽 ∧ ( 𝐽𝐵) ∈ 𝐽) → (( 𝐽𝐴) ∩ ( 𝐽𝐵)) ∈ 𝐽)
103, 6, 8, 9syl3anc 1216 . . 3 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (( 𝐽𝐴) ∩ ( 𝐽𝐵)) ∈ 𝐽)
111, 10eqeltrid 2224 . 2 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ( 𝐽 ∖ (𝐴𝐵)) ∈ 𝐽)
124cldss 12263 . . . . 5 (𝐴 ∈ (Clsd‘𝐽) → 𝐴 𝐽)
134cldss 12263 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → 𝐵 𝐽)
1412, 13anim12i 336 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 𝐽𝐵 𝐽))
15 unss 3245 . . . 4 ((𝐴 𝐽𝐵 𝐽) ↔ (𝐴𝐵) ⊆ 𝐽)
1614, 15sylib 121 . . 3 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝐵) ⊆ 𝐽)
174iscld2 12262 . . 3 ((𝐽 ∈ Top ∧ (𝐴𝐵) ⊆ 𝐽) → ((𝐴𝐵) ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ (𝐴𝐵)) ∈ 𝐽))
183, 16, 17syl2anc 408 . 2 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((𝐴𝐵) ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ (𝐴𝐵)) ∈ 𝐽))
1911, 18mpbird 166 1 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝐵) ∈ (Clsd‘𝐽))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 1480  cdif 3063  cun 3064  cin 3065  wss 3066   cuni 3731  cfv 5118  Topctop 12153  Clsdccld 12250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126  df-top 12154  df-cld 12253
This theorem is referenced by:  iuncld  12273
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