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Theorem el1o 6498
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 6491 . . 3  |-  1o  =  { (/) }
21eleq2i 2347 . 2  |-  ( A  e.  1o  <->  A  e.  {
(/) } )
3 0ex 4150 . . 3  |-  (/)  e.  _V
43elsnc2 3669 . 2  |-  ( A  e.  { (/) }  <->  A  =  (/) )
52, 4bitri 240 1  |-  ( A  e.  1o  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   (/)c0 3455   {csn 3640   1oc1o 6472
This theorem is referenced by:  0lt1o  6503  oelim2  6593  oeeulem  6599  oaabs2  6643  map0e  6805  map1  6939  cantnff  7375  cantnfreslem  7377  cnfcom3lem  7406  cfsuc  7883  pf1ind  19438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-nul 3456  df-sn 3646  df-suc 4398  df-1o 6479
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