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Theorem toponss 20544
 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 4403 . . 3 (𝐴𝐽𝐴 𝐽)
21adantl 481 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝐴 𝐽)
3 toponuni 20542 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
43adantr 480 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝑋 = 𝐽)
52, 4sseqtr4d 3605 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝐴𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ∪ cuni 4372  ‘cfv 5804  TopOnctopon 20518 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-topon 20523 This theorem is referenced by:  en2top  20600  neiptopreu  20747  iscnp3  20858  cnntr  20889  cncnp  20894  isreg2  20991  connsub  21034  iunconlem  21040  concompclo  21048  1stccnp  21075  kgenidm  21160  tx1cn  21222  tx2cn  21223  xkoccn  21232  txcnp  21233  ptcnplem  21234  xkoinjcn  21300  idqtop  21319  qtopss  21328  kqfvima  21343  kqsat  21344  kqreglem1  21354  kqreglem2  21355  qtopf1  21429  fbflim  21590  flimcf  21596  flimrest  21597  isflf  21607  fclscf  21639  subgntr  21720  ghmcnp  21728  qustgpopn  21733  qustgplem  21734  tsmsxplem1  21766  tsmsxp  21768  ressusp  21879  mopnss  22061  xrtgioo  22417  lebnumlem2  22569  cfilfcls  22880  iscmet3lem2  22898  dvres3a  23484  dvmptfsum  23542  dvcnvlem  23543  dvcnv  23544  efopn  24204  dvatan  24462  txomap  29229  cnllyscon  30481  cvmlift2lem9a  30539  icccncfext  38773  dvmptconst  38803  dvmptidg  38805  qndenserrnopnlem  39193  opnvonmbllem2  39523
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