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

Theorem ordsucg 4332
Description: The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.)
Assertion
Ref Expression
ordsucg  |-  ( A  e.  _V  ->  ( Ord  A  <->  Ord  suc  A )
)

Proof of Theorem ordsucg
StepHypRef Expression
1 ordsucim 4330 . 2  |-  ( Ord 
A  ->  Ord  suc  A
)
2 sucidg 4252 . . 3  |-  ( A  e.  _V  ->  A  e.  suc  A )
3 ordelord 4217 . . . 4  |-  ( ( Ord  suc  A  /\  A  e.  suc  A )  ->  Ord  A )
43ex 114 . . 3  |-  ( Ord 
suc  A  ->  ( A  e.  suc  A  ->  Ord  A ) )
52, 4syl5com 29 . 2  |-  ( A  e.  _V  ->  ( Ord  suc  A  ->  Ord  A ) )
61, 5impbid2 142 1  |-  ( A  e.  _V  ->  ( Ord  A  <->  Ord  suc  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    e. wcel 1439   _Vcvv 2620   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-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-3an 927  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-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-uni 3660  df-tr 3943  df-iord 4202  df-suc 4207
This theorem is referenced by:  sucelon  4333
  Copyright terms: Public domain W3C validator