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Theorem ordsssuc2 4497
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 4416 . . . . 5  |-  ( A  e.  _V  ->  ( A  e.  On  <->  Ord  A ) )
21biimprd 214 . . . 4  |-  ( A  e.  _V  ->  ( Ord  A  ->  A  e.  On ) )
32anim1d 547 . . 3  |-  ( A  e.  _V  ->  (
( Ord  A  /\  B  e.  On )  ->  ( A  e.  On  /\  B  e.  On ) ) )
4 onsssuc 4496 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  A  e.  suc  B ) )
53, 4syl6 29 . 2  |-  ( A  e.  _V  ->  (
( Ord  A  /\  B  e.  On )  ->  ( A  C_  B  <->  A  e.  suc  B ) ) )
6 annim 414 . . . . 5  |-  ( ( B  e.  On  /\  -.  A  e.  _V ) 
<->  -.  ( B  e.  On  ->  A  e.  _V ) )
7 ssexg 4176 . . . . . . 7  |-  ( ( A  C_  B  /\  B  e.  On )  ->  A  e.  _V )
87ex 423 . . . . . 6  |-  ( A 
C_  B  ->  ( B  e.  On  ->  A  e.  _V ) )
9 elex 2809 . . . . . . 7  |-  ( A  e.  suc  B  ->  A  e.  _V )
109a1d 22 . . . . . 6  |-  ( A  e.  suc  B  -> 
( B  e.  On  ->  A  e.  _V )
)
118, 10pm5.21ni 341 . . . . 5  |-  ( -.  ( B  e.  On  ->  A  e.  _V )  ->  ( A  C_  B  <->  A  e.  suc  B ) )
126, 11sylbi 187 . . . 4  |-  ( ( B  e.  On  /\  -.  A  e.  _V )  ->  ( A  C_  B 
<->  A  e.  suc  B
) )
1312expcom 424 . . 3  |-  ( -.  A  e.  _V  ->  ( B  e.  On  ->  ( A  C_  B  <->  A  e.  suc  B ) ) )
1413adantld 453 . 2  |-  ( -.  A  e.  _V  ->  ( ( Ord  A  /\  B  e.  On )  ->  ( A  C_  B  <->  A  e.  suc  B ) ) )
155, 14pm2.61i 156 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 176    /\ wa 358    e. wcel 1696   _Vcvv 2801    C_ wss 3165   Ord word 4407   Oncon0 4408   suc csuc 4410
This theorem is referenced by:  ordunisuc2  4651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414
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