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Theorem ordsucsssuc 4614
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 4613 . . . 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 4425 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
5 ordsuc 4605 . . 3  |-  ( Ord 
A  <->  Ord  suc  A )
6 ordsuc 4605 . . 3  |-  ( Ord 
B  <->  Ord  suc  B )
7 ordtri1 4425 . . 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 1684    C_ wss 3152   Ord word 4391   suc csuc 4394
This theorem is referenced by:  oawordri  6548  oeworde  6591  nnawordi  6619  bndrank  7513  ackbij1b  7865  onsuct0  24880
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398
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