Theorem eltopss 21515

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 4868	. . 3 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽)
2		1open.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
3	1, 2	sseqtrrdi 4018	. 2 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ 𝑋)
4	3	adantl 484	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ⊆ wss 3936  ∪ cuni 4838  Topctop 21501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-in 3943  df-ss 3952  df-uni 4839

This theorem is referenced by:  riinopn  21516  opncld  21641  ntrval2  21659  ntrss3  21668  cmclsopn  21670  opncldf1  21692  opnneissb  21722  opnssneib  21723  opnneiss  21726  neitr  21788  restntr  21790  cnpnei  21872  imasnopn  22298  cnextcn  22675  utopreg  22861  opnregcld  33678  ptrecube  34907  poimirlem29  34936  poimir  34940