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Theorem el1o 6334
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 6326 . . 3  |-  1o  =  { (/) }
21eleq2i 2206 . 2  |-  ( A  e.  1o  <->  A  e.  {
(/) } )
3 0ex 4055 . . 3  |-  (/)  e.  _V
43elsn2 3559 . 2  |-  ( A  e.  { (/) }  <->  A  =  (/) )
52, 4bitri 183 1  |-  ( A  e.  1o  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1331    e. wcel 1480   (/)c0 3363   {csn 3527   1oc1o 6306
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-in1 603  ax-in2 604  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-nul 4054
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-v 2688  df-dif 3073  df-un 3075  df-nul 3364  df-sn 3533  df-suc 4293  df-1o 6313
This theorem is referenced by:  0lt1o  6337  map0e  6580  map1  6706  omp1eomlem  6979  ctmlemr  6993  ctssdclemn0  6995  exmidfodomrlemeldju  7060  exmidfodomrlemreseldju  7061  1tonninf  10225
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