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Theorem ordsssuc2 4907
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 4827 . . . . 5  |-  ( A  e.  _V  ->  ( A  e.  On  <->  Ord  A ) )
21biimprd 223 . . . 4  |-  ( A  e.  _V  ->  ( Ord  A  ->  A  e.  On ) )
32anim1d 564 . . 3  |-  ( A  e.  _V  ->  (
( Ord  A  /\  B  e.  On )  ->  ( A  e.  On  /\  B  e.  On ) ) )
4 onsssuc 4906 . . 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 4538 . . . . . . 7  |-  ( ( A  C_  B  /\  B  e.  On )  ->  A  e.  _V )
87ex 434 . . . . . 6  |-  ( A 
C_  B  ->  ( B  e.  On  ->  A  e.  _V ) )
9 elex 3079 . . . . . . 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 467 . 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 1758   _Vcvv 3070    C_ wss 3428   Ord word 4818   Oncon0 4819   suc csuc 4821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-tr 4486  df-eprel 4732  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-suc 4825
This theorem is referenced by:  ordunisuc2  6557
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