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Theorem eltopss 12605
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 3814 . . 3 (𝐴𝐽𝐴 𝐽)
2 1open.1 . . 3 𝑋 = 𝐽
31, 2sseqtrrdi 3189 . 2 (𝐴𝐽𝐴𝑋)
43adantl 275 1 ((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1342  wcel 2135  wss 3114   cuni 3786  Topctop 12593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2726  df-in 3120  df-ss 3127  df-uni 3787
This theorem is referenced by:  ntrss3  12721  opnneissb  12753  opnssneib  12754  opnneiss  12756  cnpnei  12817  imasnopn  12897
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