MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ondif1 Structured version   Visualization version   Unicode version

Theorem ondif1 7221
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 7220 . 2  |-  ( A  e.  ( On  \  1o )  <->  ( A  e.  On  /\  A  =/=  (/) ) )
2 on0eln0 5485 . . 3  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
32pm5.32i 649 . 2  |-  ( ( A  e.  On  /\  (/) 
e.  A )  <->  ( A  e.  On  /\  A  =/=  (/) ) )
41, 3bitr4i 260 1  |-  ( A  e.  ( On  \  1o )  <->  ( A  e.  On  /\  (/)  e.  A
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    e. wcel 1904    =/= wne 2641    \ cdif 3387   (/)c0 3722   Oncon0 5430   1oc1o 7193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-tr 4491  df-eprel 4750  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-ord 5433  df-on 5434  df-suc 5436  df-1o 7200
This theorem is referenced by:  cantnflem2  8213  oef1o  8221  cnfcom3  8227  infxpenc  8467
  Copyright terms: Public domain W3C validator