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Theorem limsssuc 6692
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 5519 . . 3  |-  B  C_  suc  B
2 sstr2 3471 . . 3  |-  ( A 
C_  B  ->  ( B  C_  suc  B  ->  A  C_  suc  B ) )
31, 2mpi 20 . 2  |-  ( A 
C_  B  ->  A  C_ 
suc  B )
4 eleq1 2495 . . . . . . . . . . . 12  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
54biimpcd 227 . . . . . . . . . . 11  |-  ( x  e.  A  ->  (
x  =  B  ->  B  e.  A )
)
6 limsuc 6691 . . . . . . . . . . . . . 14  |-  ( Lim 
A  ->  ( B  e.  A  <->  suc  B  e.  A
) )
76biimpa 486 . . . . . . . . . . . . 13  |-  ( ( Lim  A  /\  B  e.  A )  ->  suc  B  e.  A )
8 limord 5501 . . . . . . . . . . . . . . . 16  |-  ( Lim 
A  ->  Ord  A )
98adantr 466 . . . . . . . . . . . . . . 15  |-  ( ( Lim  A  /\  B  e.  A )  ->  Ord  A )
10 ordelord 5464 . . . . . . . . . . . . . . . . 17  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )
118, 10sylan 473 . . . . . . . . . . . . . . . 16  |-  ( ( Lim  A  /\  B  e.  A )  ->  Ord  B )
12 ordsuc 6656 . . . . . . . . . . . . . . . 16  |-  ( Ord 
B  <->  Ord  suc  B )
1311, 12sylib 199 . . . . . . . . . . . . . . 15  |-  ( ( Lim  A  /\  B  e.  A )  ->  Ord  suc 
B )
14 ordtri1 5475 . . . . . . . . . . . . . . 15  |-  ( ( Ord  A  /\  Ord  suc 
B )  ->  ( A  C_  suc  B  <->  -.  suc  B  e.  A ) )
159, 13, 14syl2anc 665 . . . . . . . . . . . . . 14  |-  ( ( Lim  A  /\  B  e.  A )  ->  ( A  C_  suc  B  <->  -.  suc  B  e.  A ) )
1615con2bid 330 . . . . . . . . . . . . 13  |-  ( ( Lim  A  /\  B  e.  A )  ->  ( suc  B  e.  A  <->  -.  A  C_ 
suc  B ) )
177, 16mpbid 213 . . . . . . . . . . . 12  |-  ( ( Lim  A  /\  B  e.  A )  ->  -.  A  C_  suc  B )
1817ex 435 . . . . . . . . . . 11  |-  ( Lim 
A  ->  ( B  e.  A  ->  -.  A  C_ 
suc  B ) )
195, 18sylan9r 662 . . . . . . . . . 10  |-  ( ( Lim  A  /\  x  e.  A )  ->  (
x  =  B  ->  -.  A  C_  suc  B
) )
2019con2d 118 . . . . . . . . 9  |-  ( ( Lim  A  /\  x  e.  A )  ->  ( A  C_  suc  B  ->  -.  x  =  B
) )
2120ex 435 . . . . . . . 8  |-  ( Lim 
A  ->  ( x  e.  A  ->  ( A 
C_  suc  B  ->  -.  x  =  B ) ) )
2221com23 81 . . . . . . 7  |-  ( Lim 
A  ->  ( A  C_ 
suc  B  ->  ( x  e.  A  ->  -.  x  =  B )
) )
2322imp31 433 . . . . . 6  |-  ( ( ( Lim  A  /\  A  C_  suc  B )  /\  x  e.  A
)  ->  -.  x  =  B )
24 ssel2 3459 . . . . . . . . . 10  |-  ( ( A  C_  suc  B  /\  x  e.  A )  ->  x  e.  suc  B
)
25 vex 3083 . . . . . . . . . . 11  |-  x  e. 
_V
2625elsuc 5511 . . . . . . . . . 10  |-  ( x  e.  suc  B  <->  ( x  e.  B  \/  x  =  B ) )
2724, 26sylib 199 . . . . . . . . 9  |-  ( ( A  C_  suc  B  /\  x  e.  A )  ->  ( x  e.  B  \/  x  =  B
) )
2827ord 378 . . . . . . . 8  |-  ( ( A  C_  suc  B  /\  x  e.  A )  ->  ( -.  x  e.  B  ->  x  =  B ) )
2928con1d 127 . . . . . . 7  |-  ( ( A  C_  suc  B  /\  x  e.  A )  ->  ( -.  x  =  B  ->  x  e.  B ) )
3029adantll 718 . . . . . 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 435 . . . 4  |-  ( ( Lim  A  /\  A  C_ 
suc  B )  -> 
( x  e.  A  ->  x  e.  B ) )
3332ssrdv 3470 . . 3  |-  ( ( Lim  A  /\  A  C_ 
suc  B )  ->  A  C_  B )
3433ex 435 . 2  |-  ( Lim 
A  ->  ( A  C_ 
suc  B  ->  A  C_  B ) )
353, 34impbid2 207 1  |-  ( Lim 
A  ->  ( A  C_  B  <->  A  C_  suc  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872    C_ wss 3436   Ord word 5441   Lim wlim 5443   suc csuc 5444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660  ax-un 6598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-tr 4519  df-eprel 4764  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448
This theorem is referenced by:  cardlim  8415
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