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Theorem topopn 13968
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 3190 . . 3 𝐽𝐽
3 uniopn 13961 . . 3 ((𝐽 ∈ Top ∧ 𝐽𝐽) → 𝐽𝐽)
42, 3mpan2 425 . 2 (𝐽 ∈ Top → 𝐽𝐽)
51, 4eqeltrid 2276 1 (𝐽 ∈ Top → 𝑋𝐽)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2160  wss 3144   cuni 3824  Topctop 13957
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  ax-sep 4136
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-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592  df-uni 3825  df-top 13958
This theorem is referenced by:  toponmax  13985  cldval  14059  ntrfval  14060  clsfval  14061  iscld  14063  ntrval  14070  clsval  14071  0cld  14072  ntrtop  14088  neifval  14100  neif  14101  neival  14103  isnei  14104  tpnei  14120  cnrest  14195  txcn  14235
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