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Theorem ordsssuc2 4794
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 4714 . . . . 5  |-  ( A  e.  _V  ->  ( A  e.  On  <->  Ord  A ) )
21biimprd 223 . . . 4  |-  ( A  e.  _V  ->  ( Ord  A  ->  A  e.  On ) )
32anim1d 559 . . 3  |-  ( A  e.  _V  ->  (
( Ord  A  /\  B  e.  On )  ->  ( A  e.  On  /\  B  e.  On ) ) )
4 onsssuc 4793 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  A  e.  suc  B ) )
53, 4syl6 33 . 2  |-  ( A  e.  _V  ->  (
( Ord  A  /\  B  e.  On )  ->  ( A  C_  B  <->  A  e.  suc  B ) ) )
6 annim 425 . . . . 5  |-  ( ( B  e.  On  /\  -.  A  e.  _V ) 
<->  -.  ( B  e.  On  ->  A  e.  _V ) )
7 ssexg 4426 . . . . . . 7  |-  ( ( A  C_  B  /\  B  e.  On )  ->  A  e.  _V )
87ex 434 . . . . . 6  |-  ( A 
C_  B  ->  ( B  e.  On  ->  A  e.  _V ) )
9 elex 2971 . . . . . . 7  |-  ( A  e.  suc  B  ->  A  e.  _V )
109a1d 25 . . . . . 6  |-  ( A  e.  suc  B  -> 
( B  e.  On  ->  A  e.  _V )
)
118, 10pm5.21ni 352 . . . . 5  |-  ( -.  ( B  e.  On  ->  A  e.  _V )  ->  ( A  C_  B  <->  A  e.  suc  B ) )
126, 11sylbi 195 . . . 4  |-  ( ( B  e.  On  /\  -.  A  e.  _V )  ->  ( A  C_  B 
<->  A  e.  suc  B
) )
1312expcom 435 . . 3  |-  ( -.  A  e.  _V  ->  ( B  e.  On  ->  ( A  C_  B  <->  A  e.  suc  B ) ) )
1413adantld 464 . 2  |-  ( -.  A  e.  _V  ->  ( ( Ord  A  /\  B  e.  On )  ->  ( A  C_  B  <->  A  e.  suc  B ) ) )
155, 14pm2.61i 164 1  |-  ( ( Ord  A  /\  B  e.  On )  ->  ( A  C_  B  <->  A  e.  suc  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1755   _Vcvv 2962    C_ wss 3316   Ord word 4705   Oncon0 4706   suc csuc 4708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-tr 4374  df-eprel 4619  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-suc 4712
This theorem is referenced by:  ordunisuc2  6444
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