MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uncld Structured version   Visualization version   GIF version

Theorem uncld 21067
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 4022 . . 3 ( 𝐽 ∖ (𝐴𝐵)) = (( 𝐽𝐴) ∩ ( 𝐽𝐵))
2 cldrcl 21052 . . . . 5 (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
32adantr 472 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
4 eqid 2760 . . . . . 6 𝐽 = 𝐽
54cldopn 21057 . . . . 5 (𝐴 ∈ (Clsd‘𝐽) → ( 𝐽𝐴) ∈ 𝐽)
65adantr 472 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ( 𝐽𝐴) ∈ 𝐽)
74cldopn 21057 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → ( 𝐽𝐵) ∈ 𝐽)
87adantl 473 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ( 𝐽𝐵) ∈ 𝐽)
9 inopn 20926 . . . 4 ((𝐽 ∈ Top ∧ ( 𝐽𝐴) ∈ 𝐽 ∧ ( 𝐽𝐵) ∈ 𝐽) → (( 𝐽𝐴) ∩ ( 𝐽𝐵)) ∈ 𝐽)
103, 6, 8, 9syl3anc 1477 . . 3 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (( 𝐽𝐴) ∩ ( 𝐽𝐵)) ∈ 𝐽)
111, 10syl5eqel 2843 . 2 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ( 𝐽 ∖ (𝐴𝐵)) ∈ 𝐽)
124cldss 21055 . . . . 5 (𝐴 ∈ (Clsd‘𝐽) → 𝐴 𝐽)
134cldss 21055 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → 𝐵 𝐽)
1412, 13anim12i 591 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 𝐽𝐵 𝐽))
15 unss 3930 . . . 4 ((𝐴 𝐽𝐵 𝐽) ↔ (𝐴𝐵) ⊆ 𝐽)
1614, 15sylib 208 . . 3 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝐵) ⊆ 𝐽)
174iscld2 21054 . . 3 ((𝐽 ∈ Top ∧ (𝐴𝐵) ⊆ 𝐽) → ((𝐴𝐵) ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ (𝐴𝐵)) ∈ 𝐽))
183, 16, 17syl2anc 696 . 2 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((𝐴𝐵) ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ (𝐴𝐵)) ∈ 𝐽))
1911, 18mpbird 247 1 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝐵) ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2139  cdif 3712  cun 3713  cin 3714  wss 3715   cuni 4588  cfv 6049  Topctop 20920  Clsdccld 21042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-top 20921  df-cld 21045
This theorem is referenced by:  iscldtop  21121  paste  21320  lpcls  21390  dvasin  33827  dvacos  33828  dvreasin  33829  dvreacos  33830
  Copyright terms: Public domain W3C validator