Theorem eltopss 13594

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 3839	. . 3 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽)
2		1open.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
3	1, 2	sseqtrrdi 3206	. 2 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ 𝑋)
4	3	adantl 277	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148   ⊆ wss 3131  ∪ cuni 3811  Topctop 13582

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-ss 3144  df-uni 3812

This theorem is referenced by:  ntrss3  13708  opnneissb  13740  opnssneib  13741  opnneiss  13743  cnpnei  13804  imasnopn  13884