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Theorem toponss 20952
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 4604 . . 3 (𝐴𝐽𝐴 𝐽)
21adantl 467 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝐴 𝐽)
3 toponuni 20939 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
43adantr 466 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝑋 = 𝐽)
52, 4sseqtr4d 3791 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝐴𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wss 3723   cuni 4575  cfv 6030  TopOnctopon 20935
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-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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-ne 2944  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 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-topon 20936
This theorem is referenced by:  en2top  21010  neiptopreu  21158  iscnp3  21269  cnntr  21300  cncnp  21305  isreg2  21402  connsub  21445  iunconnlem  21451  conncompclo  21459  1stccnp  21486  kgenidm  21571  tx1cn  21633  tx2cn  21634  xkoccn  21643  txcnp  21644  ptcnplem  21645  xkoinjcn  21711  idqtop  21730  qtopss  21739  kqfvima  21754  kqsat  21755  kqreglem1  21765  kqreglem2  21766  qtopf1  21840  fbflim  22000  flimcf  22006  flimrest  22007  isflf  22017  fclscf  22049  subgntr  22130  ghmcnp  22138  qustgpopn  22143  qustgplem  22144  tsmsxplem1  22176  tsmsxp  22178  ressusp  22289  mopnss  22471  xrtgioo  22829  lebnumlem2  22981  cfilfcls  23291  iscmet3lem2  23309  dvres3a  23898  dvmptfsum  23958  dvcnvlem  23959  dvcnv  23960  efopn  24625  dvatan  24883  txomap  30241  cnllysconn  31565  cvmlift2lem9a  31623  icccncfext  40615  dvmptconst  40644  dvmptidg  40646  qndenserrnopnlem  41031  opnvonmbllem2  41364
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