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Theorem ondif1 7054
Description: Two ways to say that  A is a nonzero ordinal number. (Contributed by Mario Carneiro, 21-May-2015.)
Assertion
Ref Expression
ondif1  |-  ( A  e.  ( On  \  1o )  <->  ( A  e.  On  /\  (/)  e.  A
) )

Proof of Theorem ondif1
StepHypRef Expression
1 dif1o 7053 . 2  |-  ( A  e.  ( On  \  1o )  <->  ( A  e.  On  /\  A  =/=  (/) ) )
2 on0eln0 4885 . . 3  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
32pm5.32i 637 . 2  |-  ( ( A  e.  On  /\  (/) 
e.  A )  <->  ( A  e.  On  /\  A  =/=  (/) ) )
41, 3bitr4i 252 1  |-  ( A  e.  ( On  \  1o )  <->  ( A  e.  On  /\  (/)  e.  A
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    e. wcel 1758    =/= wne 2648    \ cdif 3436   (/)c0 3748   Oncon0 4830   1oc1o 7026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-tr 4497  df-eprel 4743  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-suc 4836  df-1o 7033
This theorem is referenced by:  cantnflem2  8012  oef1o  8044  oef1oOLD  8045  cnfcom3  8051  cnfcom3OLD  8059  infxpenc  8298  infxpencOLD  8303
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