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Theorem eltopss 11860
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 3703 . . 3 (𝐴𝐽𝐴 𝐽)
2 1open.1 . . 3 𝑋 = 𝐽
31, 2syl6sseqr 3088 . 2 (𝐴𝐽𝐴𝑋)
43adantl 272 1 ((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1296  wcel 1445  wss 3013   cuni 3675  Topctop 11848
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-in 3019  df-ss 3026  df-uni 3676
This theorem is referenced by:  ntrss3  11975  opnneissb  12007  opnssneib  12008  opnneiss  12010  cnpnei  12070
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