ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  2ordpr Unicode version

Theorem 2ordpr 4434
Description: Version of 2on 6315 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 4308 . . 3  |-  Ord  (/)
2 ordsucim 4411 . . 3  |-  ( Ord  (/)  ->  Ord  suc  (/) )
3 ordsucim 4411 . . 3  |-  ( Ord 
suc  (/)  ->  Ord  suc  suc  (/) )
41, 2, 3mp2b 8 . 2  |-  Ord  suc  suc  (/)
5 df-suc 4288 . . . 4  |-  suc  { (/)
}  =  ( {
(/) }  u.  { { (/)
} } )
6 suc0 4328 . . . . 5  |-  suc  (/)  =  { (/)
}
7 suceq 4319 . . . . 5  |-  ( suc  (/)  =  { (/) }  ->  suc 
suc  (/)  =  suc  { (/)
} )
86, 7ax-mp 5 . . . 4  |-  suc  suc  (/)  =  suc  { (/) }
9 df-pr 3529 . . . 4  |-  { (/) ,  { (/) } }  =  ( { (/) }  u.  { { (/) } } )
105, 8, 93eqtr4i 2168 . . 3  |-  suc  suc  (/)  =  { (/) ,  { (/)
} }
11 ordeq 4289 . . 3  |-  ( suc 
suc  (/)  =  { (/) ,  { (/) } }  ->  ( Ord  suc  suc  (/)  <->  Ord  { (/) ,  { (/) } } ) )
1210, 11ax-mp 5 . 2  |-  ( Ord 
suc  suc  (/)  <->  Ord  { (/) ,  { (/)
} } )
134, 12mpbi 144 1  |-  Ord  { (/)
,  { (/) } }
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1331    u. cun 3064   (/)c0 3358   {csn 3522   {cpr 3523   Ord word 4279   suc csuc 4282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-uni 3732  df-tr 4022  df-iord 4283  df-suc 4288
This theorem is referenced by:  ontr2exmid  4435  ordtri2or2exmidlem  4436  onsucelsucexmidlem  4439
  Copyright terms: Public domain W3C validator