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Theorem ordsucelsuc 6663
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 458 . . 3  |-  ( ( Ord  B  /\  A  e.  B )  ->  Ord  B )
2 ordelord 5464 . . 3  |-  ( ( Ord  B  /\  A  e.  B )  ->  Ord  A )
31, 2jca 534 . 2  |-  ( ( Ord  B  /\  A  e.  B )  ->  ( Ord  B  /\  Ord  A
) )
4 simpl 458 . . 3  |-  ( ( Ord  B  /\  suc  A  e.  suc  B )  ->  Ord  B )
5 ordsuc 6655 . . . 4  |-  ( Ord 
B  <->  Ord  suc  B )
6 ordelord 5464 . . . . 5  |-  ( ( Ord  suc  B  /\  suc  A  e.  suc  B
)  ->  Ord  suc  A
)
7 ordsuc 6655 . . . . 5  |-  ( Ord 
A  <->  Ord  suc  A )
86, 7sylibr 215 . . . 4  |-  ( ( Ord  suc  B  /\  suc  A  e.  suc  B
)  ->  Ord  A )
95, 8sylanb 474 . . 3  |-  ( ( Ord  B  /\  suc  A  e.  suc  B )  ->  Ord  A )
104, 9jca 534 . 2  |-  ( ( Ord  B  /\  suc  A  e.  suc  B )  ->  ( Ord  B  /\  Ord  A ) )
11 ordsseleq 5471 . . . . . . . 8  |-  ( ( Ord  suc  A  /\  Ord  B )  ->  ( suc  A  C_  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
127, 11sylanb 474 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  B )  ->  ( suc  A 
C_  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
1312ancoms 454 . . . . . 6  |-  ( ( Ord  B  /\  Ord  A )  ->  ( suc  A 
C_  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
1413adantl 467 . . . . 5  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( suc  A  C_  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
15 ordsucss 6659 . . . . . . 7  |-  ( Ord 
B  ->  ( A  e.  B  ->  suc  A  C_  B ) )
1615ad2antrl 732 . . . . . 6  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( A  e.  B  ->  suc 
A  C_  B )
)
17 sucssel 5534 . . . . . . 7  |-  ( A  e.  _V  ->  ( suc  A  C_  B  ->  A  e.  B ) )
1817adantr 466 . . . . . 6  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( suc  A  C_  B  ->  A  e.  B ) )
1916, 18impbid 193 . . . . 5  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( A  e.  B  <->  suc  A  C_  B ) )
20 sucexb 6650 . . . . . . 7  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
21 elsucg 5509 . . . . . . 7  |-  ( suc 
A  e.  _V  ->  ( suc  A  e.  suc  B  <-> 
( suc  A  e.  B  \/  suc  A  =  B ) ) )
2220, 21sylbi 198 . . . . . 6  |-  ( A  e.  _V  ->  ( suc  A  e.  suc  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
2322adantr 466 . . . . 5  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( suc  A  e.  suc  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
2414, 19, 233bitr4d 288 . . . 4  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( A  e.  B  <->  suc  A  e. 
suc  B ) )
2524ex 435 . . 3  |-  ( A  e.  _V  ->  (
( Ord  B  /\  Ord  A )  ->  ( A  e.  B  <->  suc  A  e. 
suc  B ) ) )
26 elex 3089 . . . . 5  |-  ( A  e.  B  ->  A  e.  _V )
27 elex 3089 . . . . . 6  |-  ( suc 
A  e.  suc  B  ->  suc  A  e.  _V )
2827, 20sylibr 215 . . . . 5  |-  ( suc 
A  e.  suc  B  ->  A  e.  _V )
2926, 28pm5.21ni 353 . . . 4  |-  ( -.  A  e.  _V  ->  ( A  e.  B  <->  suc  A  e. 
suc  B ) )
3029a1d 26 . . 3  |-  ( -.  A  e.  _V  ->  ( ( Ord  B  /\  Ord  A )  ->  ( A  e.  B  <->  suc  A  e. 
suc  B ) ) )
3125, 30pm2.61i 167 . 2  |-  ( ( Ord  B  /\  Ord  A )  ->  ( A  e.  B  <->  suc  A  e.  suc  B ) )
323, 10, 31pm5.21nd 908 1  |-  ( Ord 
B  ->  ( A  e.  B  <->  suc  A  e.  suc  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872   _Vcvv 3080    C_ wss 3436   Ord word 5441   suc csuc 5444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-tr 4519  df-eprel 4764  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-ord 5445  df-on 5446  df-suc 5448
This theorem is referenced by:  ordsucsssuc  6664  oalimcl  7272  omlimcl  7290  pssnn  7799  cantnflt  8185  cantnfp1lem3  8193  r1pw  8324  r1pwALT  8325  rankelpr  8352  rankelop  8353  rankxplim3  8360  infpssrlem4  8743  axdc3lem2  8888  axdc3lem4  8890  grur1a  9251  bnj570  29724  bnj1001  29777
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