Theorem eltopss 23045

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 ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋)

Proof of Theorem eltopss

Step	Hyp	Ref	Expression
1		elssuni 4905	. . 3 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽)
2		1open.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
3	1, 2	sseqtrrdi 3979	. 2 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ 𝑋)
4	3	adantl 486	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ⊆ wss 3906  ∪ cuni 4873  Topctop 23031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-ss 3923  df-uni 4874

This theorem is referenced by:  riinopn  23046  opncld  23171  ntrval2  23189  ntrss3  23198  cmclsopn  23200  opncldf1  23222  opnneissb  23252  opnssneib  23253  opnneiss  23256  neitr  23318  restntr  23320  cnpnei  23402  imasnopn  23828  cnextcn  24205  utopreg  24390  ist0cld  34201  opnregcld  36819  ptrecube  38249  poimirlem29  38278  poimir  38282  seposep  49681  iscnrm3rlem7  49701