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Theorem ordsucelsuc 4585
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 4386 . . 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 4577 . . . 4  |-  ( Ord 
B  <->  Ord  suc  B )
6 ordelord 4386 . . . . 5  |-  ( ( Ord  suc  B  /\  suc  A  e.  suc  B
)  ->  Ord  suc  A
)
7 ordsuc 4577 . . . . 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 4393 . . . . . . . 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 4581 . . . . . . 7  |-  ( Ord 
B  ->  ( A  e.  B  ->  suc  A  C_  B ) )
1615ad2antrl 711 . . . . . 6  |-  ( ( A  e.  _V  /\  ( Ord  B  /\  Ord  A ) )  ->  ( A  e.  B  ->  suc 
A  C_  B )
)
17 sucssel 4457 . . . . . . 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 4572 . . . . . . 7  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
21 elsucg 4431 . . . . . . 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 2771 . . . . 5  |-  ( A  e.  B  ->  A  e.  _V )
27 elex 2771 . . . . . 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 873 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 1619    e. wcel 1621   _Vcvv 2763    C_ wss 3127   Ord word 4363   suc csuc 4366
This theorem is referenced by:  ordsucsssuc  4586  oalimcl  6526  omlimcl  6544  pssnn  7049  cantnflt  7341  cantnfp1lem3  7350  r1pw  7485  r1pwOLD  7486  rankelpr  7513  rankelop  7514  rankxplim3  7519  infpssrlem4  7900  axdc3lem2  8045  axdc3lem4  8047  grur1a  8409  tartarmap  25255  bnj570  27986  bnj1001  28039
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-tr 4088  df-eprel 4277  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-suc 4370
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