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Theorem eltopss 13946
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 ∧ 𝐴𝐽) → 𝐴𝑋)

Proof of Theorem eltopss
StepHypRef Expression
1 elssuni 3852 . . 3 (𝐴𝐽𝐴 𝐽)
2 1open.1 . . 3 𝑋 = 𝐽
31, 2sseqtrrdi 3219 . 2 (𝐴𝐽𝐴𝑋)
43adantl 277 1 ((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  wss 3144   cuni 3824  Topctop 13934
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-uni 3825
This theorem is referenced by:  ntrss3  14060  opnneissb  14092  opnssneib  14093  opnneiss  14095  cnpnei  14156  imasnopn  14236
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