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Theorem ondif1 7106
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 7105 . 2  |-  ( A  e.  ( On  \  1o )  <->  ( A  e.  On  /\  A  =/=  (/) ) )
2 on0eln0 4874 . . 3  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
32pm5.32i 635 . 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 367    e. wcel 1840    =/= wne 2596    \ cdif 3408   (/)c0 3735   Oncon0 4819   1oc1o 7078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-tr 4487  df-eprel 4731  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-suc 4825  df-1o 7085
This theorem is referenced by:  cantnflem2  8059  oef1o  8091  oef1oOLD  8092  cnfcom3  8098  cnfcom3OLD  8106  infxpenc  8345  infxpencOLD  8350
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