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Theorem 2ordpr 4277
Description: Version of 2on 6040 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 4156 . . 3  |-  Ord  (/)
2 ordsucim 4254 . . 3  |-  ( Ord  (/)  ->  Ord  suc  (/) )
3 ordsucim 4254 . . 3  |-  ( Ord 
suc  (/)  ->  Ord  suc  suc  (/) )
41, 2, 3mp2b 8 . 2  |-  Ord  suc  suc  (/)
5 df-suc 4136 . . . 4  |-  suc  { (/)
}  =  ( {
(/) }  u.  { { (/)
} } )
6 suc0 4176 . . . . 5  |-  suc  (/)  =  { (/)
}
7 suceq 4167 . . . . 5  |-  ( suc  (/)  =  { (/) }  ->  suc 
suc  (/)  =  suc  { (/)
} )
86, 7ax-mp 7 . . . 4  |-  suc  suc  (/)  =  suc  { (/) }
9 df-pr 3410 . . . 4  |-  { (/) ,  { (/) } }  =  ( { (/) }  u.  { { (/) } } )
105, 8, 93eqtr4i 2086 . . 3  |-  suc  suc  (/)  =  { (/) ,  { (/)
} }
11 ordeq 4137 . . 3  |-  ( suc 
suc  (/)  =  { (/) ,  { (/) } }  ->  ( Ord  suc  suc  (/)  <->  Ord  { (/) ,  { (/) } } ) )
1210, 11ax-mp 7 . 2  |-  ( Ord 
suc  suc  (/)  <->  Ord  { (/) ,  { (/)
} } )
134, 12mpbi 137 1  |-  Ord  { (/)
,  { (/) } }
Colors of variables: wff set class
Syntax hints:    <-> wb 102    = wceq 1259    u. cun 2943   (/)c0 3252   {csn 3403   {cpr 3404   Ord word 4127   suc csuc 4130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-uni 3609  df-tr 3883  df-iord 4131  df-suc 4136
This theorem is referenced by:  ontr2exmid  4278  ordtri2or2exmidlem  4279  onsucelsucexmidlem  4282
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