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Theorem toponss 20779
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 4499 . . 3 (𝐴𝐽𝐴 𝐽)
21adantl 481 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝐴 𝐽)
3 toponuni 20767 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
43adantr 480 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝑋 = 𝐽)
52, 4sseqtr4d 3675 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝐴𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wss 3607   cuni 4468  cfv 5926  TopOnctopon 20763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-topon 20764
This theorem is referenced by:  en2top  20837  neiptopreu  20985  iscnp3  21096  cnntr  21127  cncnp  21132  isreg2  21229  connsub  21272  iunconnlem  21278  conncompclo  21286  1stccnp  21313  kgenidm  21398  tx1cn  21460  tx2cn  21461  xkoccn  21470  txcnp  21471  ptcnplem  21472  xkoinjcn  21538  idqtop  21557  qtopss  21566  kqfvima  21581  kqsat  21582  kqreglem1  21592  kqreglem2  21593  qtopf1  21667  fbflim  21827  flimcf  21833  flimrest  21834  isflf  21844  fclscf  21876  subgntr  21957  ghmcnp  21965  qustgpopn  21970  qustgplem  21971  tsmsxplem1  22003  tsmsxp  22005  ressusp  22116  mopnss  22298  xrtgioo  22656  lebnumlem2  22808  cfilfcls  23118  iscmet3lem2  23136  dvres3a  23723  dvmptfsum  23783  dvcnvlem  23784  dvcnv  23785  efopn  24449  dvatan  24707  txomap  30029  cnllysconn  31353  cvmlift2lem9a  31411  icccncfext  40418  dvmptconst  40447  dvmptidg  40449  qndenserrnopnlem  40835  opnvonmbllem2  41168
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