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Theorem ordsucelsuc 4613
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 445 . . 3  |-  ( ( Ord  B  /\  A  e.  B )  ->  Ord  B )
2 ordelord 4414 . . 3  |-  ( ( Ord  B  /\  A  e.  B )  ->  Ord  A )
31, 2jca 520 . 2  |-  ( ( Ord  B  /\  A  e.  B )  ->  ( Ord  B  /\  Ord  A
) )
4 simpl 445 . . 3  |-  ( ( Ord  B  /\  suc  A  e.  suc  B )  ->  Ord  B )
5 ordsuc 4605 . . . 4  |-  ( Ord 
B  <->  Ord  suc  B )
6 ordelord 4414 . . . . 5  |-  ( ( Ord  suc  B  /\  suc  A  e.  suc  B
)  ->  Ord  suc  A
)
7 ordsuc 4605 . . . . 5  |-  ( Ord 
A  <->  Ord  suc  A )
86, 7sylibr 205 . . . 4  |-  ( ( Ord  suc  B  /\  suc  A  e.  suc  B
)  ->  Ord  A )
95, 8sylanb 460 . . 3  |-  ( ( Ord  B  /\  suc  A  e.  suc  B )  ->  Ord  A )
104, 9jca 520 . 2  |-  ( ( Ord  B  /\  suc  A  e.  suc  B )  ->  ( Ord  B  /\  Ord  A ) )
11 ordsseleq 4421 . . . . . . . 8  |-  ( ( Ord  suc  A  /\  Ord  B )  ->  ( suc  A  C_  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
127, 11sylanb 460 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  B )  ->  ( suc  A 
C_  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
1312ancoms 441 . . . . . 6  |-  ( ( Ord  B  /\  Ord  A )  ->  ( suc  A 
C_  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
1413adantl 454 . . . . 5  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( suc  A  C_  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
15 ordsucss 4609 . . . . . . 7  |-  ( Ord 
B  ->  ( A  e.  B  ->  suc  A  C_  B ) )
1615ad2antrl 710 . . . . . 6  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( A  e.  B  ->  suc 
A  C_  B )
)
17 sucssel 4485 . . . . . . 7  |-  ( A  e.  _V  ->  ( suc  A  C_  B  ->  A  e.  B ) )
1817adantr 453 . . . . . 6  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( suc  A  C_  B  ->  A  e.  B ) )
1916, 18impbid 185 . . . . 5  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( A  e.  B  <->  suc  A  C_  B ) )
20 sucexb 4600 . . . . . . 7  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
21 elsucg 4459 . . . . . . 7  |-  ( suc 
A  e.  _V  ->  ( suc  A  e.  suc  B  <-> 
( suc  A  e.  B  \/  suc  A  =  B ) ) )
2220, 21sylbi 189 . . . . . 6  |-  ( A  e.  _V  ->  ( suc  A  e.  suc  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
2322adantr 453 . . . . 5  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( suc  A  e.  suc  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
2414, 19, 233bitr4d 278 . . . 4  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( A  e.  B  <->  suc  A  e. 
suc  B ) )
2524ex 425 . . 3  |-  ( A  e.  _V  ->  (
( Ord  B  /\  Ord  A )  ->  ( A  e.  B  <->  suc  A  e. 
suc  B ) ) )
26 elex 2798 . . . . 5  |-  ( A  e.  B  ->  A  e.  _V )
27 elex 2798 . . . . . 6  |-  ( suc 
A  e.  suc  B  ->  suc  A  e.  _V )
2827, 20sylibr 205 . . . . 5  |-  ( suc 
A  e.  suc  B  ->  A  e.  _V )
2926, 28pm5.21ni 343 . . . 4  |-  ( -.  A  e.  _V  ->  ( A  e.  B  <->  suc  A  e. 
suc  B ) )
3029a1d 24 . . 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 870 1  |-  ( Ord 
B  ->  ( A  e.  B  <->  suc  A  e.  suc  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1624    e. wcel 1685   _Vcvv 2790    C_ wss 3154   Ord word 4391   suc csuc 4394
This theorem is referenced by:  ordsucsssuc  4614  oalimcl  6554  omlimcl  6572  pssnn  7077  cantnflt  7369  cantnfp1lem3  7378  r1pw  7513  r1pwOLD  7514  rankelpr  7541  rankelop  7542  rankxplim3  7547  infpssrlem4  7928  axdc3lem2  8073  axdc3lem4  8075  grur1a  8437  tartarmap  25288  bnj570  28205  bnj1001  28258
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-tr 4116  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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