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Theorem topcld 21060
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 4095 . . . 4 (𝑋𝑋) = ∅
2 0opn 20929 . . . 4 (𝐽 ∈ Top → ∅ ∈ 𝐽)
31, 2syl5eqel 2854 . . 3 (𝐽 ∈ Top → (𝑋𝑋) ∈ 𝐽)
4 ssid 3773 . . 3 𝑋𝑋
53, 4jctil 509 . 2 (𝐽 ∈ Top → (𝑋𝑋 ∧ (𝑋𝑋) ∈ 𝐽))
6 iscld.1 . . 3 𝑋 = 𝐽
76iscld 21052 . 2 (𝐽 ∈ Top → (𝑋 ∈ (Clsd‘𝐽) ↔ (𝑋𝑋 ∧ (𝑋𝑋) ∈ 𝐽)))
85, 7mpbird 247 1 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  cdif 3720  wss 3723  c0 4063   cuni 4574  cfv 6031  Topctop 20918  Clsdccld 21041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-top 20919  df-cld 21044
This theorem is referenced by:  clsval  21062  riincld  21069  clscld  21072  clstop  21094  cldmre  21103  indiscld  21116  isconn2  21438  cnmpt2pc  22947  rlmbn  23376  ubthlem1  28066  unicls  30289  cmpfiiin  37786  kelac1  38159
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