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Theorem topopn 12189
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 3117 . . 3 𝐽𝐽
3 uniopn 12182 . . 3 ((𝐽 ∈ Top ∧ 𝐽𝐽) → 𝐽𝐽)
42, 3mpan2 421 . 2 (𝐽 ∈ Top → 𝐽𝐽)
51, 4eqeltrid 2226 1 (𝐽 ∈ Top → 𝑋𝐽)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wcel 1480  wss 3071   cuni 3736  Topctop 12178
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-uni 3737  df-top 12179
This theorem is referenced by:  toponmax  12206  cldval  12282  ntrfval  12283  clsfval  12284  iscld  12286  ntrval  12293  clsval  12294  0cld  12295  ntrtop  12311  neifval  12323  neif  12324  neival  12326  isnei  12327  tpnei  12343  cnrest  12418  txcn  12458
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