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Theorem on0eln0 5478
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 5433 . 2  |-  ( A  e.  On  ->  Ord  A )
2 ord0eln0 5477 . 2  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
31, 2syl 17 1  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    e. wcel 1887    =/= wne 2622   (/)c0 3731   Ord word 5422   Oncon0 5423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-tr 4498  df-eprel 4745  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-ord 5426  df-on 5427
This theorem is referenced by:  ondif1  7203  oe0lem  7215  oevn0  7217  oa00  7260  omord  7269  om00  7276  om00el  7277  omeulem1  7283  omeulem2  7284  oewordri  7293  oeordsuc  7295  oelim2  7296  oeoa  7298  oeoe  7300  oeeui  7303  omabs  7348  omxpenlem  7673  cantnff  8179  cantnfp1lem2  8184  cantnfp1lem3  8185  cantnfp1  8186  cantnflem1d  8193  cantnflem1  8194  cantnflem3  8196  cantnflem4  8197  cantnf  8198  cnfcomlem  8204  cnfcom3  8209  r1tskina  9207  onsucconi  31097  onint1  31109  frlmpwfi  35956
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