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Theorem topopn 11874
Description: The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
Hypothesis
Ref Expression
1open.1 𝑋 = 𝐽
Assertion
Ref Expression
topopn (𝐽 ∈ Top → 𝑋𝐽)

Proof of Theorem topopn
StepHypRef Expression
1 1open.1 . 2 𝑋 = 𝐽
2 ssid 3059 . . 3 𝐽𝐽
3 uniopn 11867 . . 3 ((𝐽 ∈ Top ∧ 𝐽𝐽) → 𝐽𝐽)
42, 3mpan2 417 . 2 (𝐽 ∈ Top → 𝐽𝐽)
51, 4syl5eqel 2181 1 (𝐽 ∈ Top → 𝑋𝐽)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1296  wcel 1445  wss 3013   cuni 3675  Topctop 11863
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  ax-sep 3978
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-ral 2375  df-rex 2376  df-v 2635  df-in 3019  df-ss 3026  df-pw 3451  df-uni 3676  df-top 11864
This theorem is referenced by:  toponmax  11890  cldval  11966  ntrfval  11967  clsfval  11968  iscld  11970  ntrval  11977  clsval  11978  0cld  11979  ntrtop  11995  neifval  12007  neif  12008  neival  12010  isnei  12011  tpnei  12027  cnrest  12101
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