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Theorem ordsucelsuc 4793
Description: Membership is inherited by successors. Generalization of Exercise 9 of [TakeutiZaring] p. 42. (Contributed by NM, 22-Jun-1998.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
Assertion
Ref Expression
ordsucelsuc  |-  ( Ord 
B  ->  ( A  e.  B  <->  suc  A  e.  suc  B ) )

Proof of Theorem ordsucelsuc
StepHypRef Expression
1 simpl 444 . . 3  |-  ( ( Ord  B  /\  A  e.  B )  ->  Ord  B )
2 ordelord 4595 . . 3  |-  ( ( Ord  B  /\  A  e.  B )  ->  Ord  A )
31, 2jca 519 . 2  |-  ( ( Ord  B  /\  A  e.  B )  ->  ( Ord  B  /\  Ord  A
) )
4 simpl 444 . . 3  |-  ( ( Ord  B  /\  suc  A  e.  suc  B )  ->  Ord  B )
5 ordsuc 4785 . . . 4  |-  ( Ord 
B  <->  Ord  suc  B )
6 ordelord 4595 . . . . 5  |-  ( ( Ord  suc  B  /\  suc  A  e.  suc  B
)  ->  Ord  suc  A
)
7 ordsuc 4785 . . . . 5  |-  ( Ord 
A  <->  Ord  suc  A )
86, 7sylibr 204 . . . 4  |-  ( ( Ord  suc  B  /\  suc  A  e.  suc  B
)  ->  Ord  A )
95, 8sylanb 459 . . 3  |-  ( ( Ord  B  /\  suc  A  e.  suc  B )  ->  Ord  A )
104, 9jca 519 . 2  |-  ( ( Ord  B  /\  suc  A  e.  suc  B )  ->  ( Ord  B  /\  Ord  A ) )
11 ordsseleq 4602 . . . . . . . 8  |-  ( ( Ord  suc  A  /\  Ord  B )  ->  ( suc  A  C_  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
127, 11sylanb 459 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  B )  ->  ( suc  A 
C_  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
1312ancoms 440 . . . . . 6  |-  ( ( Ord  B  /\  Ord  A )  ->  ( suc  A 
C_  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
1413adantl 453 . . . . 5  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( suc  A  C_  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
15 ordsucss 4789 . . . . . . 7  |-  ( Ord 
B  ->  ( A  e.  B  ->  suc  A  C_  B ) )
1615ad2antrl 709 . . . . . 6  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( A  e.  B  ->  suc 
A  C_  B )
)
17 sucssel 4665 . . . . . . 7  |-  ( A  e.  _V  ->  ( suc  A  C_  B  ->  A  e.  B ) )
1817adantr 452 . . . . . 6  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( suc  A  C_  B  ->  A  e.  B ) )
1916, 18impbid 184 . . . . 5  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( A  e.  B  <->  suc  A  C_  B ) )
20 sucexb 4780 . . . . . . 7  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
21 elsucg 4640 . . . . . . 7  |-  ( suc 
A  e.  _V  ->  ( suc  A  e.  suc  B  <-> 
( suc  A  e.  B  \/  suc  A  =  B ) ) )
2220, 21sylbi 188 . . . . . 6  |-  ( A  e.  _V  ->  ( suc  A  e.  suc  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
2322adantr 452 . . . . 5  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( suc  A  e.  suc  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
2414, 19, 233bitr4d 277 . . . 4  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( A  e.  B  <->  suc  A  e. 
suc  B ) )
2524ex 424 . . 3  |-  ( A  e.  _V  ->  (
( Ord  B  /\  Ord  A )  ->  ( A  e.  B  <->  suc  A  e. 
suc  B ) ) )
26 elex 2956 . . . . 5  |-  ( A  e.  B  ->  A  e.  _V )
27 elex 2956 . . . . . 6  |-  ( suc 
A  e.  suc  B  ->  suc  A  e.  _V )
2827, 20sylibr 204 . . . . 5  |-  ( suc 
A  e.  suc  B  ->  A  e.  _V )
2926, 28pm5.21ni 342 . . . 4  |-  ( -.  A  e.  _V  ->  ( A  e.  B  <->  suc  A  e. 
suc  B ) )
3029a1d 23 . . 3  |-  ( -.  A  e.  _V  ->  ( ( Ord  B  /\  Ord  A )  ->  ( A  e.  B  <->  suc  A  e. 
suc  B ) ) )
3125, 30pm2.61i 158 . 2  |-  ( ( Ord  B  /\  Ord  A )  ->  ( A  e.  B  <->  suc  A  e.  suc  B ) )
323, 10, 31pm5.21nd 869 1  |-  ( Ord 
B  ->  ( A  e.  B  <->  suc  A  e.  suc  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312   Ord word 4572   suc csuc 4575
This theorem is referenced by:  ordsucsssuc  4794  oalimcl  6794  omlimcl  6812  pssnn  7318  cantnflt  7616  cantnfp1lem3  7625  r1pw  7760  r1pwOLD  7761  rankelpr  7788  rankelop  7789  rankxplim3  7794  infpssrlem4  8175  axdc3lem2  8320  axdc3lem4  8322  grur1a  8683  bnj570  29130  bnj1001  29183
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 4692
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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