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Theorem toponss 22933
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 4937 . . 3 (𝐴𝐽𝐴 𝐽)
21adantl 481 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝐴 𝐽)
3 toponuni 22920 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
43adantr 480 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝑋 = 𝐽)
52, 4sseqtrrd 4021 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝐴𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wss 3951   cuni 4907  cfv 6561  TopOnctopon 22916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-topon 22917
This theorem is referenced by:  en2top  22992  neiptopreu  23141  iscnp3  23252  cnntr  23283  cncnp  23288  isreg2  23385  connsub  23429  iunconnlem  23435  conncompclo  23443  1stccnp  23470  kgenidm  23555  tx1cn  23617  tx2cn  23618  xkoccn  23627  txcnp  23628  ptcnplem  23629  xkoinjcn  23695  idqtop  23714  qtopss  23723  kqfvima  23738  kqsat  23739  kqreglem1  23749  kqreglem2  23750  qtopf1  23824  fbflim  23984  flimcf  23990  flimrest  23991  isflf  24001  fclscf  24033  subgntr  24115  ghmcnp  24123  qustgpopn  24128  qustgplem  24129  tsmsxplem1  24161  tsmsxp  24163  ressusp  24273  mopnss  24456  xrtgioo  24828  lebnumlem2  24994  cfilfcls  25308  iscmet3lem2  25326  dvres3a  25949  dvmptfsum  26013  dvcnvlem  26014  dvcnv  26015  efopn  26700  txomap  33833  cnllysconn  35250  cvmlift2lem9a  35308  icccncfext  45902  dvmptconst  45930  dvmptidg  45932  qndenserrnopnlem  46312  opnvonmbllem2  46648
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