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Theorem on0eln0 4463
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 4418 . 2  |-  ( A  e.  On  ->  Ord  A )
2 ord0eln0 4462 . 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 1696    =/= wne 2459   (/)c0 3468   Ord word 4407   Oncon0 4408
This theorem is referenced by:  ondif1  6516  oe0lem  6528  oevn0  6530  oa00  6573  omord  6582  om00  6589  om00el  6590  omeulem1  6596  omeulem2  6597  oewordri  6606  oeordsuc  6608  oelim2  6609  oeoa  6611  oeoe  6613  oeeui  6616  omabs  6661  omxpenlem  6979  cantnff  7391  cantnfp1lem2  7397  cantnfp1lem3  7398  cantnfp1  7399  cantnflem1d  7406  cantnflem1  7407  cantnflem3  7409  cantnflem4  7410  cantnf  7411  cnfcomlem  7418  cnfcom3  7423  r1tskina  8420  onsucconi  24948  onint1  24960  vtare  25988  frlmpwfi  27365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412
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