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Theorem toponss 22429
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 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝐽) β†’ 𝐴 βŠ† 𝑋)

Proof of Theorem toponss
StepHypRef Expression
1 elssuni 4942 . . 3 (𝐴 ∈ 𝐽 β†’ 𝐴 βŠ† βˆͺ 𝐽)
21adantl 483 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝐽) β†’ 𝐴 βŠ† βˆͺ 𝐽)
3 toponuni 22416 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
43adantr 482 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝐽) β†’ 𝑋 = βˆͺ 𝐽)
52, 4sseqtrrd 4024 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝐽) β†’ 𝐴 βŠ† 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   βŠ† wss 3949  βˆͺ cuni 4909  β€˜cfv 6544  TopOnctopon 22412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-topon 22413
This theorem is referenced by:  en2top  22488  neiptopreu  22637  iscnp3  22748  cnntr  22779  cncnp  22784  isreg2  22881  connsub  22925  iunconnlem  22931  conncompclo  22939  1stccnp  22966  kgenidm  23051  tx1cn  23113  tx2cn  23114  xkoccn  23123  txcnp  23124  ptcnplem  23125  xkoinjcn  23191  idqtop  23210  qtopss  23219  kqfvima  23234  kqsat  23235  kqreglem1  23245  kqreglem2  23246  qtopf1  23320  fbflim  23480  flimcf  23486  flimrest  23487  isflf  23497  fclscf  23529  subgntr  23611  ghmcnp  23619  qustgpopn  23624  qustgplem  23625  tsmsxplem1  23657  tsmsxp  23659  ressusp  23769  mopnss  23952  xrtgioo  24322  lebnumlem2  24478  cfilfcls  24791  iscmet3lem2  24809  dvres3a  25431  dvmptfsum  25492  dvcnvlem  25493  dvcnv  25494  efopn  26166  txomap  32814  cnllysconn  34236  cvmlift2lem9a  34294  icccncfext  44603  dvmptconst  44631  dvmptidg  44633  qndenserrnopnlem  45013  opnvonmbllem2  45349
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