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Theorem el1o 7141
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 7134 . . 3  |-  1o  =  { (/) }
21eleq2i 2532 . 2  |-  ( A  e.  1o  <->  A  e.  {
(/) } )
3 0ex 4569 . . 3  |-  (/)  e.  _V
43elsnc2 4047 . 2  |-  ( A  e.  { (/) }  <->  A  =  (/) )
52, 4bitri 249 1  |-  ( A  e.  1o  <->  A  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1398    e. wcel 1823   (/)c0 3783   {csn 4016   1oc1o 7115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-dif 3464  df-un 3466  df-nul 3784  df-sn 4017  df-suc 4873  df-1o 7122
This theorem is referenced by:  0lt1o  7146  oelim2  7236  oeeulem  7242  oaabs2  7286  map0e  7449  map1  7587  cantnff  8084  cnfcom3lem  8138  cnfcom3lemOLD  8146  cfsuc  8628  pf1ind  18586  mavmul0  19221  cramer0  19359
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