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Theorem topcld 22860
Description: The underlying set of a topology is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 3-Oct-2006.)
Hypothesis
Ref Expression
iscld.1 𝑋 = 𝐽
Assertion
Ref Expression
topcld (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))

Proof of Theorem topcld
StepHypRef Expression
1 difid 4370 . . . 4 (𝑋𝑋) = ∅
2 0opn 22727 . . . 4 (𝐽 ∈ Top → ∅ ∈ 𝐽)
31, 2eqeltrid 2836 . . 3 (𝐽 ∈ Top → (𝑋𝑋) ∈ 𝐽)
4 ssid 4004 . . 3 𝑋𝑋
53, 4jctil 519 . 2 (𝐽 ∈ Top → (𝑋𝑋 ∧ (𝑋𝑋) ∈ 𝐽))
6 iscld.1 . . 3 𝑋 = 𝐽
76iscld 22852 . 2 (𝐽 ∈ Top → (𝑋 ∈ (Clsd‘𝐽) ↔ (𝑋𝑋 ∧ (𝑋𝑋) ∈ 𝐽)))
85, 7mpbird 257 1 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  cdif 3945  wss 3948  c0 4322   cuni 4908  cfv 6543  Topctop 22716  Clsdccld 22841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-top 22717  df-cld 22844
This theorem is referenced by:  clsval  22862  riincld  22869  clscld  22872  clstop  22894  cldmre  22903  indiscld  22916  isconn2  23239  cnmpopc  24770  rlmbn  25210  ubthlem1  30558  unicls  33349  cmpfiiin  41901  kelac1  42271
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