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

Theorem topcld 21635
 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 4328 . . . 4 (𝑋𝑋) = ∅
2 0opn 21504 . . . 4 (𝐽 ∈ Top → ∅ ∈ 𝐽)
31, 2eqeltrid 2915 . . 3 (𝐽 ∈ Top → (𝑋𝑋) ∈ 𝐽)
4 ssid 3987 . . 3 𝑋𝑋
53, 4jctil 522 . 2 (𝐽 ∈ Top → (𝑋𝑋 ∧ (𝑋𝑋) ∈ 𝐽))
6 iscld.1 . . 3 𝑋 = 𝐽
76iscld 21627 . 2 (𝐽 ∈ Top → (𝑋 ∈ (Clsd‘𝐽) ↔ (𝑋𝑋 ∧ (𝑋𝑋) ∈ 𝐽)))
85, 7mpbird 259 1 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1530   ∈ wcel 2107   ∖ cdif 3931   ⊆ wss 3934  ∅c0 4289  ∪ cuni 4830  ‘cfv 6348  Topctop 21493  Clsdccld 21616 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-top 21494  df-cld 21619 This theorem is referenced by:  clsval  21637  riincld  21644  clscld  21647  clstop  21669  cldmre  21678  indiscld  21691  isconn2  22014  cnmpopc  23524  rlmbn  23956  ubthlem1  28639  unicls  31134  cmpfiiin  39279  kelac1  39648
 Copyright terms: Public domain W3C validator