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Theorem limsssuc 4830
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 4658 . . 3  |-  B  C_  suc  B
2 sstr2 3355 . . 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 2496 . . . . . . . . . . . 12  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
54biimpcd 216 . . . . . . . . . . 11  |-  ( x  e.  A  ->  (
x  =  B  ->  B  e.  A )
)
6 limsuc 4829 . . . . . . . . . . . . . 14  |-  ( Lim 
A  ->  ( B  e.  A  <->  suc  B  e.  A
) )
76biimpa 471 . . . . . . . . . . . . 13  |-  ( ( Lim  A  /\  B  e.  A )  ->  suc  B  e.  A )
8 limord 4640 . . . . . . . . . . . . . . . 16  |-  ( Lim 
A  ->  Ord  A )
98adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( Lim  A  /\  B  e.  A )  ->  Ord  A )
10 ordelord 4603 . . . . . . . . . . . . . . . . 17  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )
118, 10sylan 458 . . . . . . . . . . . . . . . 16  |-  ( ( Lim  A  /\  B  e.  A )  ->  Ord  B )
12 ordsuc 4794 . . . . . . . . . . . . . . . 16  |-  ( Ord 
B  <->  Ord  suc  B )
1311, 12sylib 189 . . . . . . . . . . . . . . 15  |-  ( ( Lim  A  /\  B  e.  A )  ->  Ord  suc 
B )
14 ordtri1 4614 . . . . . . . . . . . . . . 15  |-  ( ( Ord  A  /\  Ord  suc 
B )  ->  ( A  C_  suc  B  <->  -.  suc  B  e.  A ) )
159, 13, 14syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( Lim  A  /\  B  e.  A )  ->  ( A  C_  suc  B  <->  -.  suc  B  e.  A ) )
1615con2bid 320 . . . . . . . . . . . . 13  |-  ( ( Lim  A  /\  B  e.  A )  ->  ( suc  B  e.  A  <->  -.  A  C_ 
suc  B ) )
177, 16mpbid 202 . . . . . . . . . . . 12  |-  ( ( Lim  A  /\  B  e.  A )  ->  -.  A  C_  suc  B )
1817ex 424 . . . . . . . . . . 11  |-  ( Lim 
A  ->  ( B  e.  A  ->  -.  A  C_ 
suc  B ) )
195, 18sylan9r 640 . . . . . . . . . 10  |-  ( ( Lim  A  /\  x  e.  A )  ->  (
x  =  B  ->  -.  A  C_  suc  B
) )
2019con2d 109 . . . . . . . . 9  |-  ( ( Lim  A  /\  x  e.  A )  ->  ( A  C_  suc  B  ->  -.  x  =  B
) )
2120ex 424 . . . . . . . 8  |-  ( Lim 
A  ->  ( x  e.  A  ->  ( A 
C_  suc  B  ->  -.  x  =  B ) ) )
2221com23 74 . . . . . . 7  |-  ( Lim 
A  ->  ( A  C_ 
suc  B  ->  ( x  e.  A  ->  -.  x  =  B )
) )
2322imp31 422 . . . . . 6  |-  ( ( ( Lim  A  /\  A  C_  suc  B )  /\  x  e.  A
)  ->  -.  x  =  B )
24 ssel2 3343 . . . . . . . . . 10  |-  ( ( A  C_  suc  B  /\  x  e.  A )  ->  x  e.  suc  B
)
25 vex 2959 . . . . . . . . . . 11  |-  x  e. 
_V
2625elsuc 4650 . . . . . . . . . 10  |-  ( x  e.  suc  B  <->  ( x  e.  B  \/  x  =  B ) )
2724, 26sylib 189 . . . . . . . . 9  |-  ( ( A  C_  suc  B  /\  x  e.  A )  ->  ( x  e.  B  \/  x  =  B
) )
2827ord 367 . . . . . . . 8  |-  ( ( A  C_  suc  B  /\  x  e.  A )  ->  ( -.  x  e.  B  ->  x  =  B ) )
2928con1d 118 . . . . . . 7  |-  ( ( A  C_  suc  B  /\  x  e.  A )  ->  ( -.  x  =  B  ->  x  e.  B ) )
3029adantll 695 . . . . . 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 424 . . . 4  |-  ( ( Lim  A  /\  A  C_ 
suc  B )  -> 
( x  e.  A  ->  x  e.  B ) )
3332ssrdv 3354 . . 3  |-  ( ( Lim  A  /\  A  C_ 
suc  B )  ->  A  C_  B )
3433ex 424 . 2  |-  ( Lim 
A  ->  ( A  C_ 
suc  B  ->  A  C_  B ) )
353, 34impbid2 196 1  |-  ( Lim 
A  ->  ( A  C_  B  <->  A  C_  suc  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3320   Ord word 4580   Lim wlim 4582   suc csuc 4583
This theorem is referenced by:  cardlim  7859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587
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