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Theorem ordsucsssuc 4616
Description: The subclass relationship between two ordinal classes is inherited by their successors. (Contributed by NM, 4-Oct-2003.)
Assertion
Ref Expression
ordsucsssuc  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  suc  A  C_  suc  B ) )

Proof of Theorem ordsucsssuc
StepHypRef Expression
1 ordsucelsuc 4615 . . . 4  |-  ( Ord 
A  ->  ( B  e.  A  <->  suc  B  e.  suc  A ) )
21notbid 285 . . 3  |-  ( Ord 
A  ->  ( -.  B  e.  A  <->  -.  suc  B  e.  suc  A ) )
32adantr 451 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  B  e.  A  <->  -.  suc  B  e.  suc  A ) )
4 ordtri1 4427 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
5 ordsuc 4607 . . 3  |-  ( Ord 
A  <->  Ord  suc  A )
6 ordsuc 4607 . . 3  |-  ( Ord 
B  <->  Ord  suc  B )
7 ordtri1 4427 . . 3  |-  ( ( Ord  suc  A  /\  Ord  suc  B )  -> 
( suc  A  C_  suc  B  <->  -.  suc  B  e.  suc  A ) )
85, 6, 7syl2anb 465 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( suc  A 
C_  suc  B  <->  -.  suc  B  e.  suc  A ) )
93, 4, 83bitr4d 276 1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  suc  A  C_  suc  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1686    C_ wss 3154   Ord word 4393   suc csuc 4396
This theorem is referenced by:  oawordri  6550  oeworde  6593  nnawordi  6621  bndrank  7515  ackbij1b  7867  onsuct0  24882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-tr 4116  df-eprel 4307  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-suc 4400
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