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Theorem on0eln0 4447
Description: An ordinal number contains zero iff it is nonzero. (Contributed by NM, 6-Dec-2004.)
Assertion
Ref Expression
on0eln0  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )

Proof of Theorem on0eln0
StepHypRef Expression
1 eloni 4402 . 2  |-  ( A  e.  On  ->  Ord  A )
2 ord0eln0 4446 . 2  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
31, 2syl 15 1  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1684    =/= wne 2446   (/)c0 3455   Ord word 4391   Oncon0 4392
This theorem is referenced by:  ondif1  6500  oe0lem  6512  oevn0  6514  oa00  6557  omord  6566  om00  6573  om00el  6574  omeulem1  6580  omeulem2  6581  oewordri  6590  oeordsuc  6592  oelim2  6593  oeoa  6595  oeoe  6597  oeeui  6600  omabs  6645  omxpenlem  6963  cantnff  7375  cantnfp1lem2  7381  cantnfp1lem3  7382  cantnfp1  7383  cantnflem1d  7390  cantnflem1  7391  cantnflem3  7393  cantnflem4  7394  cantnf  7395  cnfcomlem  7402  cnfcom3  7407  r1tskina  8404  onsucconi  24876  onint1  24888  vtare  25885  frlmpwfi  27262
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
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