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Theorem toponss 21463
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 4859 . . 3 (𝐴𝐽𝐴 𝐽)
21adantl 482 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝐴 𝐽)
3 toponuni 21450 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
43adantr 481 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝑋 = 𝐽)
52, 4sseqtrrd 4005 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝐴𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wss 3933   cuni 4830  cfv 6348  TopOnctopon 21446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  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-topon 21447
This theorem is referenced by:  en2top  21521  neiptopreu  21669  iscnp3  21780  cnntr  21811  cncnp  21816  isreg2  21913  connsub  21957  iunconnlem  21963  conncompclo  21971  1stccnp  21998  kgenidm  22083  tx1cn  22145  tx2cn  22146  xkoccn  22155  txcnp  22156  ptcnplem  22157  xkoinjcn  22223  idqtop  22242  qtopss  22251  kqfvima  22266  kqsat  22267  kqreglem1  22277  kqreglem2  22278  qtopf1  22352  fbflim  22512  flimcf  22518  flimrest  22519  isflf  22529  fclscf  22561  subgntr  22642  ghmcnp  22650  qustgpopn  22655  qustgplem  22656  tsmsxplem1  22688  tsmsxp  22690  ressusp  22801  mopnss  22983  xrtgioo  23341  lebnumlem2  23493  cfilfcls  23804  iscmet3lem2  23822  dvres3a  24439  dvmptfsum  24499  dvcnvlem  24500  dvcnv  24501  efopn  25168  txomap  30997  cnllysconn  32389  cvmlift2lem9a  32447  icccncfext  42046  dvmptconst  42075  dvmptidg  42077  qndenserrnopnlem  42459  opnvonmbllem2  42792
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