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Theorem 2ordpr 4353
Description: Version of 2on 6204 with the definition of  2o expanded and expressed in terms of  Ord. (Contributed by Jim Kingdon, 29-Aug-2021.)
Assertion
Ref Expression
2ordpr  |-  Ord  { (/)
,  { (/) } }

Proof of Theorem 2ordpr
StepHypRef Expression
1 ord0 4227 . . 3  |-  Ord  (/)
2 ordsucim 4330 . . 3  |-  ( Ord  (/)  ->  Ord  suc  (/) )
3 ordsucim 4330 . . 3  |-  ( Ord 
suc  (/)  ->  Ord  suc  suc  (/) )
41, 2, 3mp2b 8 . 2  |-  Ord  suc  suc  (/)
5 df-suc 4207 . . . 4  |-  suc  { (/)
}  =  ( {
(/) }  u.  { { (/)
} } )
6 suc0 4247 . . . . 5  |-  suc  (/)  =  { (/)
}
7 suceq 4238 . . . . 5  |-  ( suc  (/)  =  { (/) }  ->  suc 
suc  (/)  =  suc  { (/)
} )
86, 7ax-mp 7 . . . 4  |-  suc  suc  (/)  =  suc  { (/) }
9 df-pr 3457 . . . 4  |-  { (/) ,  { (/) } }  =  ( { (/) }  u.  { { (/) } } )
105, 8, 93eqtr4i 2119 . . 3  |-  suc  suc  (/)  =  { (/) ,  { (/)
} }
11 ordeq 4208 . . 3  |-  ( suc 
suc  (/)  =  { (/) ,  { (/) } }  ->  ( Ord  suc  suc  (/)  <->  Ord  { (/) ,  { (/) } } ) )
1210, 11ax-mp 7 . 2  |-  ( Ord 
suc  suc  (/)  <->  Ord  { (/) ,  { (/)
} } )
134, 12mpbi 144 1  |-  Ord  { (/)
,  { (/) } }
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1290    u. cun 2998   (/)c0 3287   {csn 3450   {cpr 3451   Ord word 4198   suc csuc 4201
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-uni 3660  df-tr 3943  df-iord 4202  df-suc 4207
This theorem is referenced by:  ontr2exmid  4354  ordtri2or2exmidlem  4355  onsucelsucexmidlem  4358
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