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Theorem el1o 6745
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 6738 . . 3  |-  1o  =  { (/) }
21eleq2i 2502 . 2  |-  ( A  e.  1o  <->  A  e.  {
(/) } )
3 0ex 4341 . . 3  |-  (/)  e.  _V
43elsnc2 3845 . 2  |-  ( A  e.  { (/) }  <->  A  =  (/) )
52, 4bitri 242 1  |-  ( A  e.  1o  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1726   (/)c0 3630   {csn 3816   1oc1o 6719
This theorem is referenced by:  0lt1o  6750  oelim2  6840  oeeulem  6846  oaabs2  6890  map0e  7053  map1  7187  cantnff  7631  cantnfreslem  7633  cnfcom3lem  7662  cfsuc  8139  pf1ind  19977
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-nul 4340
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-un 3327  df-nul 3631  df-sn 3822  df-suc 4589  df-1o 6726
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