Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  eltopss Structured version   Visualization version   GIF version

Theorem eltopss 21490
 Description: A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)
Hypothesis
Ref Expression
1open.1 𝑋 = 𝐽
Assertion
Ref Expression
eltopss ((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝑋)

Proof of Theorem eltopss
StepHypRef Expression
1 elssuni 4841 . . 3 (𝐴𝐽𝐴 𝐽)
2 1open.1 . . 3 𝑋 = 𝐽
31, 2sseqtrrdi 3994 . 2 (𝐴𝐽𝐴𝑋)
43adantl 485 1 ((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115   ⊆ wss 3910  ∪ cuni 4811  Topctop 21476 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-v 3473  df-in 3917  df-ss 3927  df-uni 4812 This theorem is referenced by:  riinopn  21491  opncld  21616  ntrval2  21634  ntrss3  21643  cmclsopn  21645  opncldf1  21667  opnneissb  21697  opnssneib  21698  opnneiss  21701  neitr  21763  restntr  21765  cnpnei  21847  imasnopn  22273  cnextcn  22650  utopreg  22836  opnregcld  33685  ptrecube  34937  poimirlem29  34966  poimir  34970
 Copyright terms: Public domain W3C validator