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Theorem ordsssuc2 4662
Description: An ordinal subset of an ordinal number belongs to its successor. (Contributed by NM, 1-Feb-2005.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
Assertion
Ref Expression
ordsssuc2  |-  ( ( Ord  A  /\  B  e.  On )  ->  ( A  C_  B  <->  A  e.  suc  B ) )

Proof of Theorem ordsssuc2
StepHypRef Expression
1 elong 4581 . . . . 5  |-  ( A  e.  _V  ->  ( A  e.  On  <->  Ord  A ) )
21biimprd 215 . . . 4  |-  ( A  e.  _V  ->  ( Ord  A  ->  A  e.  On ) )
32anim1d 548 . . 3  |-  ( A  e.  _V  ->  (
( Ord  A  /\  B  e.  On )  ->  ( A  e.  On  /\  B  e.  On ) ) )
4 onsssuc 4661 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  A  e.  suc  B ) )
53, 4syl6 31 . 2  |-  ( A  e.  _V  ->  (
( Ord  A  /\  B  e.  On )  ->  ( A  C_  B  <->  A  e.  suc  B ) ) )
6 annim 415 . . . . 5  |-  ( ( B  e.  On  /\  -.  A  e.  _V ) 
<->  -.  ( B  e.  On  ->  A  e.  _V ) )
7 ssexg 4341 . . . . . . 7  |-  ( ( A  C_  B  /\  B  e.  On )  ->  A  e.  _V )
87ex 424 . . . . . 6  |-  ( A 
C_  B  ->  ( B  e.  On  ->  A  e.  _V ) )
9 elex 2956 . . . . . . 7  |-  ( A  e.  suc  B  ->  A  e.  _V )
109a1d 23 . . . . . 6  |-  ( A  e.  suc  B  -> 
( B  e.  On  ->  A  e.  _V )
)
118, 10pm5.21ni 342 . . . . 5  |-  ( -.  ( B  e.  On  ->  A  e.  _V )  ->  ( A  C_  B  <->  A  e.  suc  B ) )
126, 11sylbi 188 . . . 4  |-  ( ( B  e.  On  /\  -.  A  e.  _V )  ->  ( A  C_  B 
<->  A  e.  suc  B
) )
1312expcom 425 . . 3  |-  ( -.  A  e.  _V  ->  ( B  e.  On  ->  ( A  C_  B  <->  A  e.  suc  B ) ) )
1413adantld 454 . 2  |-  ( -.  A  e.  _V  ->  ( ( Ord  A  /\  B  e.  On )  ->  ( A  C_  B  <->  A  e.  suc  B ) ) )
155, 14pm2.61i 158 1  |-  ( ( Ord  A  /\  B  e.  On )  ->  ( A  C_  B  <->  A  e.  suc  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725   _Vcvv 2948    C_ wss 3312   Ord word 4572   Oncon0 4573   suc csuc 4575
This theorem is referenced by:  ordunisuc2  4816
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-suc 4579
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