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

Theorem topopn 20931
Description: The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
Hypothesis
Ref Expression
1open.1 𝑋 = 𝐽
Assertion
Ref Expression
topopn (𝐽 ∈ Top → 𝑋𝐽)

Proof of Theorem topopn
StepHypRef Expression
1 1open.1 . 2 𝑋 = 𝐽
2 ssid 3773 . . 3 𝐽𝐽
3 uniopn 20922 . . 3 ((𝐽 ∈ Top ∧ 𝐽𝐽) → 𝐽𝐽)
42, 3mpan2 671 . 2 (𝐽 ∈ Top → 𝐽𝐽)
51, 4syl5eqel 2854 1 (𝐽 ∈ Top → 𝑋𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  wss 3723   cuni 4574  Topctop 20918
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
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-in 3730  df-ss 3737  df-pw 4299  df-uni 4575  df-top 20919
This theorem is referenced by:  riinopn  20933  toponmax  20951  cldval  21048  ntrfval  21049  clsfval  21050  iscld  21052  ntrval  21061  clsval  21062  0cld  21063  clsval2  21075  ntrtop  21095  toponmre  21118  neifval  21124  neif  21125  neival  21127  isnei  21128  tpnei  21146  lpfval  21163  lpval  21164  restcld  21197  restcls  21206  restntr  21207  cnrest  21310  cmpsub  21424  hauscmplem  21430  cmpfi  21432  isconn2  21438  connsubclo  21448  1stcfb  21469  1stcelcls  21485  islly2  21508  lly1stc  21520  islocfin  21541  finlocfin  21544  cmpkgen  21575  llycmpkgen  21576  ptbasid  21599  ptpjpre2  21604  ptopn2  21608  xkoopn  21613  xkouni  21623  txcld  21627  txcn  21650  ptrescn  21663  txtube  21664  txhaus  21671  xkoptsub  21678  xkopt  21679  xkopjcn  21680  qtoptop  21724  qtopuni  21726  opnfbas  21866  flimval  21987  flimfil  21993  hausflim  22005  hauspwpwf1  22011  hauspwpwdom  22012  flimfnfcls  22052  cnpfcfi  22064  bcthlem5  23344  dvply1  24259  cldssbrsiga  30590  dya2iocucvr  30686  kur14lem7  31532  kur14lem9  31534  connpconn  31555  cvmliftmolem1  31601  ordtop  32772  ntrelmap  38949  clselmap  38951  dssmapntrcls  38952  dssmapclsntr  38953  reopn  40019
  Copyright terms: Public domain W3C validator