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Theorem ord0eln0 5494
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 3768 . 2  |-  ( (/)  e.  A  ->  A  =/=  (/) )
2 df-ne 2621 . . 3  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
3 ord0 5492 . . . . 5  |-  Ord  (/)
4 noel 3766 . . . . . 6  |-  -.  A  e.  (/)
5 ordtri2 5475 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  (/) )  ->  ( A  e.  (/)  <->  -.  ( A  =  (/)  \/  (/)  e.  A
) ) )
65con2bid 331 . . . . . 6  |-  ( ( Ord  A  /\  Ord  (/) )  ->  ( ( A  =  (/)  \/  (/)  e.  A
)  <->  -.  A  e.  (/) ) )
74, 6mpbiri 237 . . . . 5  |-  ( ( Ord  A  /\  Ord  (/) )  ->  ( A  =  (/)  \/  (/)  e.  A
) )
83, 7mpan2 676 . . . 4  |-  ( Ord 
A  ->  ( A  =  (/)  \/  (/)  e.  A
) )
98ord 379 . . 3  |-  ( Ord 
A  ->  ( -.  A  =  (/)  ->  (/)  e.  A
) )
102, 9syl5bi 221 . 2  |-  ( Ord 
A  ->  ( A  =/=  (/)  ->  (/)  e.  A
) )
111, 10impbid2 208 1  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    = wceq 1438    e. wcel 1869    =/= wne 2619   (/)c0 3762   Ord word 5439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-tr 4517  df-eprel 4762  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-ord 5443
This theorem is referenced by:  on0eln0  5495  dflim2  5496  0ellim  5502  0elsuc  6674  ordge1n0  7206  omwordi  7278  omass  7287  nnmord  7339  nnmwordi  7342  wemapwe  8205  elni2  9304  bnj529  29553
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