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Theorem on0eln0 4638
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 4593 . 2  |-  ( A  e.  On  ->  Ord  A )
2 ord0eln0 4637 . 2  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
31, 2syl 16 1  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    e. wcel 1726    =/= wne 2601   (/)c0 3630   Ord word 4582   Oncon0 4583
This theorem is referenced by:  ondif1  6747  oe0lem  6759  oevn0  6761  oa00  6804  omord  6813  om00  6820  om00el  6821  omeulem1  6827  omeulem2  6828  oewordri  6837  oeordsuc  6839  oelim2  6840  oeoa  6842  oeoe  6844  oeeui  6847  omabs  6892  omxpenlem  7211  cantnff  7631  cantnfp1lem2  7637  cantnfp1lem3  7638  cantnfp1  7639  cantnflem1d  7646  cantnflem1  7647  cantnflem3  7649  cantnflem4  7650  cantnf  7651  cnfcomlem  7658  cnfcom3  7663  r1tskina  8659  onsucconi  26189  onint1  26201  frlmpwfi  27241
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-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587
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