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Theorem limsssuc 4641
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 4469 . . 3  |-  B  C_  suc  B
2 sstr2 3186 . . 3  |-  ( A 
C_  B  ->  ( B  C_  suc  B  ->  A  C_  suc  B ) )
31, 2mpi 16 . 2  |-  ( A 
C_  B  ->  A  C_ 
suc  B )
4 eleq1 2343 . . . . . . . . . . . 12  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
54biimpcd 215 . . . . . . . . . . 11  |-  ( x  e.  A  ->  (
x  =  B  ->  B  e.  A )
)
6 limsuc 4640 . . . . . . . . . . . . . 14  |-  ( Lim 
A  ->  ( B  e.  A  <->  suc  B  e.  A
) )
76biimpa 470 . . . . . . . . . . . . 13  |-  ( ( Lim  A  /\  B  e.  A )  ->  suc  B  e.  A )
8 limord 4451 . . . . . . . . . . . . . . . 16  |-  ( Lim 
A  ->  Ord  A )
98adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( Lim  A  /\  B  e.  A )  ->  Ord  A )
10 ordelord 4414 . . . . . . . . . . . . . . . . 17  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )
118, 10sylan 457 . . . . . . . . . . . . . . . 16  |-  ( ( Lim  A  /\  B  e.  A )  ->  Ord  B )
12 ordsuc 4605 . . . . . . . . . . . . . . . 16  |-  ( Ord 
B  <->  Ord  suc  B )
1311, 12sylib 188 . . . . . . . . . . . . . . 15  |-  ( ( Lim  A  /\  B  e.  A )  ->  Ord  suc 
B )
14 ordtri1 4425 . . . . . . . . . . . . . . 15  |-  ( ( Ord  A  /\  Ord  suc 
B )  ->  ( A  C_  suc  B  <->  -.  suc  B  e.  A ) )
159, 13, 14syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( Lim  A  /\  B  e.  A )  ->  ( A  C_  suc  B  <->  -.  suc  B  e.  A ) )
1615con2bid 319 . . . . . . . . . . . . 13  |-  ( ( Lim  A  /\  B  e.  A )  ->  ( suc  B  e.  A  <->  -.  A  C_ 
suc  B ) )
177, 16mpbid 201 . . . . . . . . . . . 12  |-  ( ( Lim  A  /\  B  e.  A )  ->  -.  A  C_  suc  B )
1817ex 423 . . . . . . . . . . 11  |-  ( Lim 
A  ->  ( B  e.  A  ->  -.  A  C_ 
suc  B ) )
195, 18sylan9r 639 . . . . . . . . . 10  |-  ( ( Lim  A  /\  x  e.  A )  ->  (
x  =  B  ->  -.  A  C_  suc  B
) )
2019con2d 107 . . . . . . . . 9  |-  ( ( Lim  A  /\  x  e.  A )  ->  ( A  C_  suc  B  ->  -.  x  =  B
) )
2120ex 423 . . . . . . . 8  |-  ( Lim 
A  ->  ( x  e.  A  ->  ( A 
C_  suc  B  ->  -.  x  =  B ) ) )
2221com23 72 . . . . . . 7  |-  ( Lim 
A  ->  ( A  C_ 
suc  B  ->  ( x  e.  A  ->  -.  x  =  B )
) )
2322imp31 421 . . . . . 6  |-  ( ( ( Lim  A  /\  A  C_  suc  B )  /\  x  e.  A
)  ->  -.  x  =  B )
24 ssel2 3175 . . . . . . . . . 10  |-  ( ( A  C_  suc  B  /\  x  e.  A )  ->  x  e.  suc  B
)
25 vex 2791 . . . . . . . . . . 11  |-  x  e. 
_V
2625elsuc 4461 . . . . . . . . . 10  |-  ( x  e.  suc  B  <->  ( x  e.  B  \/  x  =  B ) )
2724, 26sylib 188 . . . . . . . . 9  |-  ( ( A  C_  suc  B  /\  x  e.  A )  ->  ( x  e.  B  \/  x  =  B
) )
2827ord 366 . . . . . . . 8  |-  ( ( A  C_  suc  B  /\  x  e.  A )  ->  ( -.  x  e.  B  ->  x  =  B ) )
2928con1d 116 . . . . . . 7  |-  ( ( A  C_  suc  B  /\  x  e.  A )  ->  ( -.  x  =  B  ->  x  e.  B ) )
3029adantll 694 . . . . . 6  |-  ( ( ( Lim  A  /\  A  C_  suc  B )  /\  x  e.  A
)  ->  ( -.  x  =  B  ->  x  e.  B ) )
3123, 30mpd 14 . . . . 5  |-  ( ( ( Lim  A  /\  A  C_  suc  B )  /\  x  e.  A
)  ->  x  e.  B )
3231ex 423 . . . 4  |-  ( ( Lim  A  /\  A  C_ 
suc  B )  -> 
( x  e.  A  ->  x  e.  B ) )
3332ssrdv 3185 . . 3  |-  ( ( Lim  A  /\  A  C_ 
suc  B )  ->  A  C_  B )
3433ex 423 . 2  |-  ( Lim 
A  ->  ( A  C_ 
suc  B  ->  A  C_  B ) )
353, 34impbid2 195 1  |-  ( Lim 
A  ->  ( A  C_  B  <->  A  C_  suc  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   Ord word 4391   Lim wlim 4393   suc csuc 4394
This theorem is referenced by:  cardlim  7605
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398
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