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Theorem el1o 7050
Description: Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
Assertion
Ref Expression
el1o  |-  ( A  e.  1o  <->  A  =  (/) )

Proof of Theorem el1o
StepHypRef Expression
1 df1o2 7043 . . 3  |-  1o  =  { (/) }
21eleq2i 2532 . 2  |-  ( A  e.  1o  <->  A  e.  {
(/) } )
3 0ex 4531 . . 3  |-  (/)  e.  _V
43elsnc2 4017 . 2  |-  ( A  e.  { (/) }  <->  A  =  (/) )
52, 4bitri 249 1  |-  ( A  e.  1o  <->  A  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1370    e. wcel 1758   (/)c0 3746   {csn 3986   1oc1o 7024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-nul 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-v 3080  df-dif 3440  df-un 3442  df-nul 3747  df-sn 3987  df-suc 4834  df-1o 7031
This theorem is referenced by:  0lt1o  7055  oelim2  7145  oeeulem  7151  oaabs2  7195  map0e  7361  map1  7499  cantnff  7994  cnfcom3lem  8048  cnfcom3lemOLD  8056  cfsuc  8538  pf1ind  17915  mavmul0  18491  cramer0  18629
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