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Theorem uncld 11965
 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 3267 . . 3 ( 𝐽 ∖ (𝐴𝐵)) = (( 𝐽𝐴) ∩ ( 𝐽𝐵))
2 cldrcl 11954 . . . . 5 (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
32adantr 271 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
4 eqid 2095 . . . . . 6 𝐽 = 𝐽
54cldopn 11959 . . . . 5 (𝐴 ∈ (Clsd‘𝐽) → ( 𝐽𝐴) ∈ 𝐽)
65adantr 271 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ( 𝐽𝐴) ∈ 𝐽)
74cldopn 11959 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → ( 𝐽𝐵) ∈ 𝐽)
87adantl 272 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ( 𝐽𝐵) ∈ 𝐽)
9 inopn 11854 . . . 4 ((𝐽 ∈ Top ∧ ( 𝐽𝐴) ∈ 𝐽 ∧ ( 𝐽𝐵) ∈ 𝐽) → (( 𝐽𝐴) ∩ ( 𝐽𝐵)) ∈ 𝐽)
103, 6, 8, 9syl3anc 1181 . . 3 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (( 𝐽𝐴) ∩ ( 𝐽𝐵)) ∈ 𝐽)
111, 10syl5eqel 2181 . 2 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ( 𝐽 ∖ (𝐴𝐵)) ∈ 𝐽)
124cldss 11957 . . . . 5 (𝐴 ∈ (Clsd‘𝐽) → 𝐴 𝐽)
134cldss 11957 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → 𝐵 𝐽)
1412, 13anim12i 332 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 𝐽𝐵 𝐽))
15 unss 3189 . . . 4 ((𝐴 𝐽𝐵 𝐽) ↔ (𝐴𝐵) ⊆ 𝐽)
1614, 15sylib 121 . . 3 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝐵) ⊆ 𝐽)
174iscld2 11956 . . 3 ((𝐽 ∈ Top ∧ (𝐴𝐵) ⊆ 𝐽) → ((𝐴𝐵) ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ (𝐴𝐵)) ∈ 𝐽))
183, 16, 17syl2anc 404 . 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 1445   ∖ cdif 3010   ∪ cun 3011   ∩ cin 3012   ⊆ wss 3013  ∪ cuni 3675  ‘cfv 5049  Topctop 11848  Clsdccld 11944 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284 This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fn 5052  df-fv 5057  df-top 11849  df-cld 11947 This theorem is referenced by:  iuncld  11967
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