Theorem eltopss 14525

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 3880	. . 3 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽)
2		1open.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
3	1, 2	sseqtrrdi 3243	. 2 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ 𝑋)
4	3	adantl 277	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177   ⊆ wss 3167  ∪ cuni 3852  Topctop 14513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3173  df-ss 3180  df-uni 3853

This theorem is referenced by:  ntrss3  14639  opnneissb  14671  opnssneib  14672  opnneiss  14674  cnpnei  14735  imasnopn  14815