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Theorem 2ordpr 4330
Description: Version of 2on 6172 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 4209 . . 3  |-  Ord  (/)
2 ordsucim 4307 . . 3  |-  ( Ord  (/)  ->  Ord  suc  (/) )
3 ordsucim 4307 . . 3  |-  ( Ord 
suc  (/)  ->  Ord  suc  suc  (/) )
41, 2, 3mp2b 8 . 2  |-  Ord  suc  suc  (/)
5 df-suc 4189 . . . 4  |-  suc  { (/)
}  =  ( {
(/) }  u.  { { (/)
} } )
6 suc0 4229 . . . . 5  |-  suc  (/)  =  { (/)
}
7 suceq 4220 . . . . 5  |-  ( suc  (/)  =  { (/) }  ->  suc 
suc  (/)  =  suc  { (/)
} )
86, 7ax-mp 7 . . . 4  |-  suc  suc  (/)  =  suc  { (/) }
9 df-pr 3448 . . . 4  |-  { (/) ,  { (/) } }  =  ( { (/) }  u.  { { (/) } } )
105, 8, 93eqtr4i 2118 . . 3  |-  suc  suc  (/)  =  { (/) ,  { (/)
} }
11 ordeq 4190 . . 3  |-  ( suc 
suc  (/)  =  { (/) ,  { (/) } }  ->  ( Ord  suc  suc  (/)  <->  Ord  { (/) ,  { (/) } } ) )
1210, 11ax-mp 7 . 2  |-  ( Ord 
suc  suc  (/)  <->  Ord  { (/) ,  { (/)
} } )
134, 12mpbi 143 1  |-  Ord  { (/)
,  { (/) } }
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1289    u. cun 2995   (/)c0 3284   {csn 3441   {cpr 3442   Ord word 4180   suc csuc 4183
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-uni 3649  df-tr 3929  df-iord 4184  df-suc 4189
This theorem is referenced by:  ontr2exmid  4331  ordtri2or2exmidlem  4332  onsucelsucexmidlem  4335
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