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Theorem limsssuc 6663
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 4955 . . 3  |-  B  C_  suc  B
2 sstr2 3511 . . 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 2539 . . . . . . . . . . . 12  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
54biimpcd 224 . . . . . . . . . . 11  |-  ( x  e.  A  ->  (
x  =  B  ->  B  e.  A )
)
6 limsuc 6662 . . . . . . . . . . . . . 14  |-  ( Lim 
A  ->  ( B  e.  A  <->  suc  B  e.  A
) )
76biimpa 484 . . . . . . . . . . . . 13  |-  ( ( Lim  A  /\  B  e.  A )  ->  suc  B  e.  A )
8 limord 4937 . . . . . . . . . . . . . . . 16  |-  ( Lim 
A  ->  Ord  A )
98adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( Lim  A  /\  B  e.  A )  ->  Ord  A )
10 ordelord 4900 . . . . . . . . . . . . . . . . 17  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )
118, 10sylan 471 . . . . . . . . . . . . . . . 16  |-  ( ( Lim  A  /\  B  e.  A )  ->  Ord  B )
12 ordsuc 6627 . . . . . . . . . . . . . . . 16  |-  ( Ord 
B  <->  Ord  suc  B )
1311, 12sylib 196 . . . . . . . . . . . . . . 15  |-  ( ( Lim  A  /\  B  e.  A )  ->  Ord  suc 
B )
14 ordtri1 4911 . . . . . . . . . . . . . . 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 3499 . . . . . . . . . 10  |-  ( ( A  C_  suc  B  /\  x  e.  A )  ->  x  e.  suc  B
)
25 vex 3116 . . . . . . . . . . 11  |-  x  e. 
_V
2625elsuc 4947 . . . . . . . . . 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 3510 . . 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 1379    e. wcel 1767    C_ wss 3476   Ord word 4877   Lim wlim 4879   suc csuc 4880
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 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884
This theorem is referenced by:  cardlim  8349
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