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Theorem ord0eln0 4599
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 3598 . 2  |-  ( (/)  e.  A  ->  A  =/=  (/) )
2 df-ne 2573 . . 3  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
3 ord0 4597 . . . . 5  |-  Ord  (/)
4 noel 3596 . . . . . 6  |-  -.  A  e.  (/)
5 ordtri2 4580 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  (/) )  ->  ( A  e.  (/)  <->  -.  ( A  =  (/)  \/  (/)  e.  A
) ) )
65con2bid 320 . . . . . 6  |-  ( ( Ord  A  /\  Ord  (/) )  ->  ( ( A  =  (/)  \/  (/)  e.  A
)  <->  -.  A  e.  (/) ) )
74, 6mpbiri 225 . . . . 5  |-  ( ( Ord  A  /\  Ord  (/) )  ->  ( A  =  (/)  \/  (/)  e.  A
) )
83, 7mpan2 653 . . . 4  |-  ( Ord 
A  ->  ( A  =  (/)  \/  (/)  e.  A
) )
98ord 367 . . 3  |-  ( Ord 
A  ->  ( -.  A  =  (/)  ->  (/)  e.  A
) )
102, 9syl5bi 209 . 2  |-  ( Ord 
A  ->  ( A  =/=  (/)  ->  (/)  e.  A
) )
111, 10impbid2 196 1  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2571   (/)c0 3592   Ord word 4544
This theorem is referenced by:  on0eln0  4600  dflim2  4601  0ellim  4607  0elsuc  4778  ordge1n0  6705  omwordi  6777  omass  6786  nnmord  6838  nnmwordi  6841  wemapwe  7614  elni2  8714  bnj529  28819
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-tr 4267  df-eprel 4458  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548
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