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Theorem ordsucsssuc 4795
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 4794 . . . 4  |-  ( Ord 
A  ->  ( B  e.  A  <->  suc  B  e.  suc  A ) )
21notbid 286 . . 3  |-  ( Ord 
A  ->  ( -.  B  e.  A  <->  -.  suc  B  e.  suc  A ) )
32adantr 452 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  B  e.  A  <->  -.  suc  B  e.  suc  A ) )
4 ordtri1 4606 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
5 ordsuc 4786 . . 3  |-  ( Ord 
A  <->  Ord  suc  A )
6 ordsuc 4786 . . 3  |-  ( Ord 
B  <->  Ord  suc  B )
7 ordtri1 4606 . . 3  |-  ( ( Ord  suc  A  /\  Ord  suc  B )  -> 
( suc  A  C_  suc  B  <->  -.  suc  B  e.  suc  A ) )
85, 6, 7syl2anb 466 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( suc  A 
C_  suc  B  <->  -.  suc  B  e.  suc  A ) )
93, 4, 83bitr4d 277 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 177    /\ wa 359    e. wcel 1725    C_ wss 3312   Ord word 4572   suc csuc 4575
This theorem is referenced by:  oawordri  6785  oeworde  6828  nnawordi  6856  bndrank  7759  ackbij1b  8111  onsuct0  26183
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-suc 4579
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