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

Theorem toponss 21536
 Description: A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
toponss ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝐴𝑋)

Proof of Theorem toponss
StepHypRef Expression
1 elssuni 4833 . . 3 (𝐴𝐽𝐴 𝐽)
21adantl 485 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝐴 𝐽)
3 toponuni 21523 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
43adantr 484 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝑋 = 𝐽)
52, 4sseqtrrd 3959 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝐴𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112   ⊆ wss 3884  ∪ cuni 4803  ‘cfv 6328  TopOnctopon 21519 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-topon 21520 This theorem is referenced by:  en2top  21594  neiptopreu  21742  iscnp3  21853  cnntr  21884  cncnp  21889  isreg2  21986  connsub  22030  iunconnlem  22036  conncompclo  22044  1stccnp  22071  kgenidm  22156  tx1cn  22218  tx2cn  22219  xkoccn  22228  txcnp  22229  ptcnplem  22230  xkoinjcn  22296  idqtop  22315  qtopss  22324  kqfvima  22339  kqsat  22340  kqreglem1  22350  kqreglem2  22351  qtopf1  22425  fbflim  22585  flimcf  22591  flimrest  22592  isflf  22602  fclscf  22634  subgntr  22716  ghmcnp  22724  qustgpopn  22729  qustgplem  22730  tsmsxplem1  22762  tsmsxp  22764  ressusp  22875  mopnss  23057  xrtgioo  23415  lebnumlem2  23571  cfilfcls  23882  iscmet3lem2  23900  dvres3a  24521  dvmptfsum  24582  dvcnvlem  24583  dvcnv  24584  efopn  25253  txomap  31191  cnllysconn  32606  cvmlift2lem9a  32664  icccncfext  42522  dvmptconst  42550  dvmptidg  42552  qndenserrnopnlem  42932  opnvonmbllem2  43265
 Copyright terms: Public domain W3C validator