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Theorem limsssuc 6558
Description: A class includes a limit ordinal iff the successor of the class includes it. (Contributed by NM, 30-Oct-2003.)
Assertion
Ref Expression
limsssuc  |-  ( Lim 
A  ->  ( A  C_  B  <->  A  C_  suc  B
) )

Proof of Theorem limsssuc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sssucid 4891 . . 3  |-  B  C_  suc  B
2 sstr2 3458 . . 3  |-  ( A 
C_  B  ->  ( B  C_  suc  B  ->  A  C_  suc  B ) )
31, 2mpi 17 . 2  |-  ( A 
C_  B  ->  A  C_ 
suc  B )
4 eleq1 2521 . . . . . . . . . . . 12  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
54biimpcd 224 . . . . . . . . . . 11  |-  ( x  e.  A  ->  (
x  =  B  ->  B  e.  A )
)
6 limsuc 6557 . . . . . . . . . . . . . 14  |-  ( Lim 
A  ->  ( B  e.  A  <->  suc  B  e.  A
) )
76biimpa 484 . . . . . . . . . . . . 13  |-  ( ( Lim  A  /\  B  e.  A )  ->  suc  B  e.  A )
8 limord 4873 . . . . . . . . . . . . . . . 16  |-  ( Lim 
A  ->  Ord  A )
98adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( Lim  A  /\  B  e.  A )  ->  Ord  A )
10 ordelord 4836 . . . . . . . . . . . . . . . . 17  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )
118, 10sylan 471 . . . . . . . . . . . . . . . 16  |-  ( ( Lim  A  /\  B  e.  A )  ->  Ord  B )
12 ordsuc 6522 . . . . . . . . . . . . . . . 16  |-  ( Ord 
B  <->  Ord  suc  B )
1311, 12sylib 196 . . . . . . . . . . . . . . 15  |-  ( ( Lim  A  /\  B  e.  A )  ->  Ord  suc 
B )
14 ordtri1 4847 . . . . . . . . . . . . . . 15  |-  ( ( Ord  A  /\  Ord  suc 
B )  ->  ( A  C_  suc  B  <->  -.  suc  B  e.  A ) )
159, 13, 14syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( Lim  A  /\  B  e.  A )  ->  ( A  C_  suc  B  <->  -.  suc  B  e.  A ) )
1615con2bid 329 . . . . . . . . . . . . 13  |-  ( ( Lim  A  /\  B  e.  A )  ->  ( suc  B  e.  A  <->  -.  A  C_ 
suc  B ) )
177, 16mpbid 210 . . . . . . . . . . . 12  |-  ( ( Lim  A  /\  B  e.  A )  ->  -.  A  C_  suc  B )
1817ex 434 . . . . . . . . . . 11  |-  ( Lim 
A  ->  ( B  e.  A  ->  -.  A  C_ 
suc  B ) )
195, 18sylan9r 658 . . . . . . . . . 10  |-  ( ( Lim  A  /\  x  e.  A )  ->  (
x  =  B  ->  -.  A  C_  suc  B
) )
2019con2d 115 . . . . . . . . 9  |-  ( ( Lim  A  /\  x  e.  A )  ->  ( A  C_  suc  B  ->  -.  x  =  B
) )
2120ex 434 . . . . . . . 8  |-  ( Lim 
A  ->  ( x  e.  A  ->  ( A 
C_  suc  B  ->  -.  x  =  B ) ) )
2221com23 78 . . . . . . 7  |-  ( Lim 
A  ->  ( A  C_ 
suc  B  ->  ( x  e.  A  ->  -.  x  =  B )
) )
2322imp31 432 . . . . . 6  |-  ( ( ( Lim  A  /\  A  C_  suc  B )  /\  x  e.  A
)  ->  -.  x  =  B )
24 ssel2 3446 . . . . . . . . . 10  |-  ( ( A  C_  suc  B  /\  x  e.  A )  ->  x  e.  suc  B
)
25 vex 3068 . . . . . . . . . . 11  |-  x  e. 
_V
2625elsuc 4883 . . . . . . . . . 10  |-  ( x  e.  suc  B  <->  ( x  e.  B  \/  x  =  B ) )
2724, 26sylib 196 . . . . . . . . 9  |-  ( ( A  C_  suc  B  /\  x  e.  A )  ->  ( x  e.  B  \/  x  =  B
) )
2827ord 377 . . . . . . . 8  |-  ( ( A  C_  suc  B  /\  x  e.  A )  ->  ( -.  x  e.  B  ->  x  =  B ) )
2928con1d 124 . . . . . . 7  |-  ( ( A  C_  suc  B  /\  x  e.  A )  ->  ( -.  x  =  B  ->  x  e.  B ) )
3029adantll 713 . . . . . 6  |-  ( ( ( Lim  A  /\  A  C_  suc  B )  /\  x  e.  A
)  ->  ( -.  x  =  B  ->  x  e.  B ) )
3123, 30mpd 15 . . . . 5  |-  ( ( ( Lim  A  /\  A  C_  suc  B )  /\  x  e.  A
)  ->  x  e.  B )
3231ex 434 . . . 4  |-  ( ( Lim  A  /\  A  C_ 
suc  B )  -> 
( x  e.  A  ->  x  e.  B ) )
3332ssrdv 3457 . . 3  |-  ( ( Lim  A  /\  A  C_ 
suc  B )  ->  A  C_  B )
3433ex 434 . 2  |-  ( Lim 
A  ->  ( A  C_ 
suc  B  ->  A  C_  B ) )
353, 34impbid2 204 1  |-  ( Lim 
A  ->  ( A  C_  B  <->  A  C_  suc  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    C_ wss 3423   Ord word 4813   Lim wlim 4815   suc csuc 4816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-tr 4481  df-eprel 4727  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820
This theorem is referenced by:  cardlim  8240
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