MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordsucelsuc Structured version   Unicode version

Theorem ordsucelsuc 6652
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 457 . . 3  |-  ( ( Ord  B  /\  A  e.  B )  ->  Ord  B )
2 ordelord 4906 . . 3  |-  ( ( Ord  B  /\  A  e.  B )  ->  Ord  A )
31, 2jca 532 . 2  |-  ( ( Ord  B  /\  A  e.  B )  ->  ( Ord  B  /\  Ord  A
) )
4 simpl 457 . . 3  |-  ( ( Ord  B  /\  suc  A  e.  suc  B )  ->  Ord  B )
5 ordsuc 6644 . . . 4  |-  ( Ord 
B  <->  Ord  suc  B )
6 ordelord 4906 . . . . 5  |-  ( ( Ord  suc  B  /\  suc  A  e.  suc  B
)  ->  Ord  suc  A
)
7 ordsuc 6644 . . . . 5  |-  ( Ord 
A  <->  Ord  suc  A )
86, 7sylibr 212 . . . 4  |-  ( ( Ord  suc  B  /\  suc  A  e.  suc  B
)  ->  Ord  A )
95, 8sylanb 472 . . 3  |-  ( ( Ord  B  /\  suc  A  e.  suc  B )  ->  Ord  A )
104, 9jca 532 . 2  |-  ( ( Ord  B  /\  suc  A  e.  suc  B )  ->  ( Ord  B  /\  Ord  A ) )
11 ordsseleq 4913 . . . . . . . 8  |-  ( ( Ord  suc  A  /\  Ord  B )  ->  ( suc  A  C_  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
127, 11sylanb 472 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  B )  ->  ( suc  A 
C_  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
1312ancoms 453 . . . . . 6  |-  ( ( Ord  B  /\  Ord  A )  ->  ( suc  A 
C_  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
1413adantl 466 . . . . 5  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( suc  A  C_  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
15 ordsucss 6648 . . . . . . 7  |-  ( Ord 
B  ->  ( A  e.  B  ->  suc  A  C_  B ) )
1615ad2antrl 727 . . . . . 6  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( A  e.  B  ->  suc 
A  C_  B )
)
17 sucssel 4976 . . . . . . 7  |-  ( A  e.  _V  ->  ( suc  A  C_  B  ->  A  e.  B ) )
1817adantr 465 . . . . . 6  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( suc  A  C_  B  ->  A  e.  B ) )
1916, 18impbid 191 . . . . 5  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( A  e.  B  <->  suc  A  C_  B ) )
20 sucexb 6639 . . . . . . 7  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
21 elsucg 4951 . . . . . . 7  |-  ( suc 
A  e.  _V  ->  ( suc  A  e.  suc  B  <-> 
( suc  A  e.  B  \/  suc  A  =  B ) ) )
2220, 21sylbi 195 . . . . . 6  |-  ( A  e.  _V  ->  ( suc  A  e.  suc  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
2322adantr 465 . . . . 5  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( suc  A  e.  suc  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
2414, 19, 233bitr4d 285 . . . 4  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( A  e.  B  <->  suc  A  e. 
suc  B ) )
2524ex 434 . . 3  |-  ( A  e.  _V  ->  (
( Ord  B  /\  Ord  A )  ->  ( A  e.  B  <->  suc  A  e. 
suc  B ) ) )
26 elex 3127 . . . . 5  |-  ( A  e.  B  ->  A  e.  _V )
27 elex 3127 . . . . . 6  |-  ( suc 
A  e.  suc  B  ->  suc  A  e.  _V )
2827, 20sylibr 212 . . . . 5  |-  ( suc 
A  e.  suc  B  ->  A  e.  _V )
2926, 28pm5.21ni 352 . . . 4  |-  ( -.  A  e.  _V  ->  ( A  e.  B  <->  suc  A  e. 
suc  B ) )
3029a1d 25 . . 3  |-  ( -.  A  e.  _V  ->  ( ( Ord  B  /\  Ord  A )  ->  ( A  e.  B  <->  suc  A  e. 
suc  B ) ) )
3125, 30pm2.61i 164 . 2  |-  ( ( Ord  B  /\  Ord  A )  ->  ( A  e.  B  <->  suc  A  e.  suc  B ) )
323, 10, 31pm5.21nd 898 1  |-  ( Ord 
B  ->  ( A  e.  B  <->  suc  A  e.  suc  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    C_ wss 3481   Ord word 4883   suc csuc 4886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-tr 4547  df-eprel 4797  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-suc 4890
This theorem is referenced by:  ordsucsssuc  6653  oalimcl  7221  omlimcl  7239  pssnn  7750  cantnflt  8103  cantnfp1lem3  8111  cantnfltOLD  8133  cantnfp1lem3OLD  8137  r1pw  8275  r1pwOLD  8276  rankelpr  8303  rankelop  8304  rankxplim3  8311  infpssrlem4  8698  axdc3lem2  8843  axdc3lem4  8845  grur1a  9209  bnj570  33448  bnj1001  33501
  Copyright terms: Public domain W3C validator