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Theorem uncld 12907
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 3379 . . 3 ( 𝐽 ∖ (𝐴𝐵)) = (( 𝐽𝐴) ∩ ( 𝐽𝐵))
2 cldrcl 12896 . . . . 5 (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
32adantr 274 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
4 eqid 2170 . . . . . 6 𝐽 = 𝐽
54cldopn 12901 . . . . 5 (𝐴 ∈ (Clsd‘𝐽) → ( 𝐽𝐴) ∈ 𝐽)
65adantr 274 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ( 𝐽𝐴) ∈ 𝐽)
74cldopn 12901 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → ( 𝐽𝐵) ∈ 𝐽)
87adantl 275 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ( 𝐽𝐵) ∈ 𝐽)
9 inopn 12795 . . . 4 ((𝐽 ∈ Top ∧ ( 𝐽𝐴) ∈ 𝐽 ∧ ( 𝐽𝐵) ∈ 𝐽) → (( 𝐽𝐴) ∩ ( 𝐽𝐵)) ∈ 𝐽)
103, 6, 8, 9syl3anc 1233 . . 3 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (( 𝐽𝐴) ∩ ( 𝐽𝐵)) ∈ 𝐽)
111, 10eqeltrid 2257 . 2 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ( 𝐽 ∖ (𝐴𝐵)) ∈ 𝐽)
124cldss 12899 . . . . 5 (𝐴 ∈ (Clsd‘𝐽) → 𝐴 𝐽)
134cldss 12899 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → 𝐵 𝐽)
1412, 13anim12i 336 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 𝐽𝐵 𝐽))
15 unss 3301 . . . 4 ((𝐴 𝐽𝐵 𝐽) ↔ (𝐴𝐵) ⊆ 𝐽)
1614, 15sylib 121 . . 3 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝐵) ⊆ 𝐽)
174iscld2 12898 . . 3 ((𝐽 ∈ Top ∧ (𝐴𝐵) ⊆ 𝐽) → ((𝐴𝐵) ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ (𝐴𝐵)) ∈ 𝐽))
183, 16, 17syl2anc 409 . 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 2141  cdif 3118  cun 3119  cin 3120  wss 3121   cuni 3796  cfv 5198  Topctop 12789  Clsdccld 12886
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206  df-top 12790  df-cld 12889
This theorem is referenced by:  iuncld  12909
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