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Theorem ordsucsssuc 6537
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 6536 . . . 4  |-  ( Ord 
A  ->  ( B  e.  A  <->  suc  B  e.  suc  A ) )
21notbid 294 . . 3  |-  ( Ord 
A  ->  ( -.  B  e.  A  <->  -.  suc  B  e.  suc  A ) )
32adantr 465 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  B  e.  A  <->  -.  suc  B  e.  suc  A ) )
4 ordtri1 4853 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
5 ordsuc 6528 . . 3  |-  ( Ord 
A  <->  Ord  suc  A )
6 ordsuc 6528 . . 3  |-  ( Ord 
B  <->  Ord  suc  B )
7 ordtri1 4853 . . 3  |-  ( ( Ord  suc  A  /\  Ord  suc  B )  -> 
( suc  A  C_  suc  B  <->  -.  suc  B  e.  suc  A ) )
85, 6, 7syl2anb 479 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( suc  A 
C_  suc  B  <->  -.  suc  B  e.  suc  A ) )
93, 4, 83bitr4d 285 1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  suc  A  C_  suc  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1758    C_ wss 3429   Ord word 4819   suc csuc 4822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-tr 4487  df-eprel 4733  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-suc 4826
This theorem is referenced by:  oawordri  7092  oeworde  7135  nnawordi  7163  bndrank  8152  rankmapu  8189  ackbij1b  8512  onsuct0  28424
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