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Theorem topcld 20591
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 3901 . . . 4 (𝑋𝑋) = ∅
2 0opn 20476 . . . 4 (𝐽 ∈ Top → ∅ ∈ 𝐽)
31, 2syl5eqel 2691 . . 3 (𝐽 ∈ Top → (𝑋𝑋) ∈ 𝐽)
4 ssid 3586 . . 3 𝑋𝑋
53, 4jctil 557 . 2 (𝐽 ∈ Top → (𝑋𝑋 ∧ (𝑋𝑋) ∈ 𝐽))
6 iscld.1 . . 3 𝑋 = 𝐽
76iscld 20583 . 2 (𝐽 ∈ Top → (𝑋 ∈ (Clsd‘𝐽) ↔ (𝑋𝑋 ∧ (𝑋𝑋) ∈ 𝐽)))
85, 7mpbird 245 1 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  cdif 3536  wss 3539  c0 3873   cuni 4366  cfv 5790  Topctop 20459  Clsdccld 20572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-top 20463  df-cld 20575
This theorem is referenced by:  clsval  20593  riincld  20600  clscld  20603  clstop  20625  cldmre  20634  indiscld  20647  iscon2  20969  cnmpt2pc  22466  rlmbn  22882  ubthlem1  26916  unicls  29083  cmpfiiin  36074  kelac1  36447
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