Theorem eltopss 22872

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 4881	. . 3 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽)
2		1open.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
3	1, 2	sseqtrrdi 3963	. 2 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ 𝑋)
4	3	adantl 481	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3889  ∪ cuni 4850  Topctop 22858

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-ss 3906  df-uni 4851

This theorem is referenced by:  riinopn  22873  opncld  22998  ntrval2  23016  ntrss3  23025  cmclsopn  23027  opncldf1  23049  opnneissb  23079  opnssneib  23080  opnneiss  23083  neitr  23145  restntr  23147  cnpnei  23229  imasnopn  23655  cnextcn  24032  utopreg  24217  ist0cld  33977  opnregcld  36512  ptrecube  37941  poimirlem29  37970  poimir  37974  seposep  49401  iscnrm3rlem7  49421