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Theorem ordsucsssuc 6643
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 6642 . . . 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 4901 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
5 ordsuc 6634 . . 3  |-  ( Ord 
A  <->  Ord  suc  A )
6 ordsuc 6634 . . 3  |-  ( Ord 
B  <->  Ord  suc  B )
7 ordtri1 4901 . . 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 1804    C_ wss 3461   Ord word 4867   suc csuc 4870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-tr 4531  df-eprel 4781  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-suc 4874
This theorem is referenced by:  oawordri  7201  oeworde  7244  nnawordi  7272  bndrank  8262  rankmapu  8299  ackbij1b  8622  onsuct0  29881
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