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Theorem ord0eln0 4549
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 3549 . 2  |-  ( (/)  e.  A  ->  A  =/=  (/) )
2 df-ne 2531 . . 3  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
3 ord0 4547 . . . . 5  |-  Ord  (/)
4 noel 3547 . . . . . 6  |-  -.  A  e.  (/)
5 ordtri2 4530 . . . . . . 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 1647    e. wcel 1715    =/= wne 2529   (/)c0 3543   Ord word 4494
This theorem is referenced by:  on0eln0  4550  dflim2  4551  0ellim  4557  0elsuc  4729  ordge1n0  6639  omwordi  6711  omass  6720  nnmord  6772  nnmwordi  6775  wemapwe  7547  elni2  8648  bnj529  28534
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-tr 4216  df-eprel 4408  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498
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