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Theorem ord0eln0 4446
Description: A non-empty 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 3461 . 2  |-  ( (/)  e.  A  ->  A  =/=  (/) )
2 df-ne 2448 . . 3  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
3 ord0 4444 . . . . 5  |-  Ord  (/)
4 noel 3459 . . . . . 6  |-  -.  A  e.  (/)
5 ordtri2 4427 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  (/) )  ->  ( A  e.  (/)  <->  -.  ( A  =  (/)  \/  (/)  e.  A
) ) )
65con2bid 319 . . . . . 6  |-  ( ( Ord  A  /\  Ord  (/) )  ->  ( ( A  =  (/)  \/  (/)  e.  A
)  <->  -.  A  e.  (/) ) )
74, 6mpbiri 224 . . . . 5  |-  ( ( Ord  A  /\  Ord  (/) )  ->  ( A  =  (/)  \/  (/)  e.  A
) )
83, 7mpan2 652 . . . 4  |-  ( Ord 
A  ->  ( A  =  (/)  \/  (/)  e.  A
) )
98ord 366 . . 3  |-  ( Ord 
A  ->  ( -.  A  =  (/)  ->  (/)  e.  A
) )
102, 9syl5bi 208 . 2  |-  ( Ord 
A  ->  ( A  =/=  (/)  ->  (/)  e.  A
) )
111, 10impbid2 195 1  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   (/)c0 3455   Ord word 4391
This theorem is referenced by:  on0eln0  4447  dflim2  4448  0ellim  4454  0elsuc  4626  ordge1n0  6497  omwordi  6569  omass  6578  nnmord  6630  nnmwordi  6633  wemapwe  7400  elni2  8501  bnj529  28770
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395
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