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Theorem topcld 23043
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 4376 . . . 4 (𝑋𝑋) = ∅
2 0opn 22910 . . . 4 (𝐽 ∈ Top → ∅ ∈ 𝐽)
31, 2eqeltrid 2845 . . 3 (𝐽 ∈ Top → (𝑋𝑋) ∈ 𝐽)
4 ssid 4006 . . 3 𝑋𝑋
53, 4jctil 519 . 2 (𝐽 ∈ Top → (𝑋𝑋 ∧ (𝑋𝑋) ∈ 𝐽))
6 iscld.1 . . 3 𝑋 = 𝐽
76iscld 23035 . 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 2108  cdif 3948  wss 3951  c0 4333   cuni 4907  cfv 6561  Topctop 22899  Clsdccld 23024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-top 22900  df-cld 23027
This theorem is referenced by:  clsval  23045  riincld  23052  clscld  23055  clstop  23077  cldmre  23086  indiscld  23099  isconn2  23422  cnmpopc  24955  rlmbn  25395  ubthlem1  30889  unicls  33902  cmpfiiin  42708  kelac1  43075
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