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Theorem ord0eln0 4871
Description: A nonempty ordinal contains the empty set. (Contributed by NM, 25-Nov-1995.)
Assertion
Ref Expression
ord0eln0  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )

Proof of Theorem ord0eln0
StepHypRef Expression
1 ne0i 3741 . 2  |-  ( (/)  e.  A  ->  A  =/=  (/) )
2 df-ne 2646 . . 3  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
3 ord0 4869 . . . . 5  |-  Ord  (/)
4 noel 3739 . . . . . 6  |-  -.  A  e.  (/)
5 ordtri2 4852 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  (/) )  ->  ( A  e.  (/)  <->  -.  ( A  =  (/)  \/  (/)  e.  A
) ) )
65con2bid 329 . . . . . 6  |-  ( ( Ord  A  /\  Ord  (/) )  ->  ( ( A  =  (/)  \/  (/)  e.  A
)  <->  -.  A  e.  (/) ) )
74, 6mpbiri 233 . . . . 5  |-  ( ( Ord  A  /\  Ord  (/) )  ->  ( A  =  (/)  \/  (/)  e.  A
) )
83, 7mpan2 671 . . . 4  |-  ( Ord 
A  ->  ( A  =  (/)  \/  (/)  e.  A
) )
98ord 377 . . 3  |-  ( Ord 
A  ->  ( -.  A  =  (/)  ->  (/)  e.  A
) )
102, 9syl5bi 217 . 2  |-  ( Ord 
A  ->  ( A  =/=  (/)  ->  (/)  e.  A
) )
111, 10impbid2 204 1  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   (/)c0 3735   Ord word 4816
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 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-tr 4484  df-eprel 4730  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820
This theorem is referenced by:  on0eln0  4872  dflim2  4873  0ellim  4879  0elsuc  6546  ordge1n0  7038  omwordi  7110  omass  7119  nnmord  7171  nnmwordi  7174  wemapwe  8029  wemapweOLD  8030  elni2  9147  bnj529  32033
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