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Theorem fisucdomOLD 7066
Description: Strict dominance of a finite set over a natural number is the same as dominance over its successor. (Contributed by NM, 26-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
fisucdomOLD  |-  ( ( A  e.  om  /\  B  e.  Fin )  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) )

Proof of Theorem fisucdomOLD
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 isfi 6885 . 2  |-  ( B  e.  Fin  <->  E. x  e.  om  B  ~~  x
)
2 omsucdomOLD 7056 . . . . . . 7  |-  ( ( A  e.  om  /\  x  e.  om )  ->  ( A  ~<  x  <->  suc 
A  ~<_  x ) )
32adantr 451 . . . . . 6  |-  ( ( ( A  e.  om  /\  x  e.  om )  /\  B  ~~  x )  ->  ( A  ~<  x  <->  suc  A  ~<_  x ) )
4 sdomen2 7006 . . . . . . 7  |-  ( B 
~~  x  ->  ( A  ~<  B  <->  A  ~<  x ) )
54adantl 452 . . . . . 6  |-  ( ( ( A  e.  om  /\  x  e.  om )  /\  B  ~~  x )  ->  ( A  ~<  B  <-> 
A  ~<  x ) )
6 domen2 7004 . . . . . . 7  |-  ( B 
~~  x  ->  ( suc  A  ~<_  B  <->  suc  A  ~<_  x ) )
76adantl 452 . . . . . 6  |-  ( ( ( A  e.  om  /\  x  e.  om )  /\  B  ~~  x )  ->  ( suc  A  ~<_  B 
<->  suc  A  ~<_  x ) )
83, 5, 73bitr4d 276 . . . . 5  |-  ( ( ( A  e.  om  /\  x  e.  om )  /\  B  ~~  x )  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) )
98exp31 587 . . . 4  |-  ( A  e.  om  ->  (
x  e.  om  ->  ( B  ~~  x  -> 
( A  ~<  B  <->  suc  A  ~<_  B ) ) ) )
109rexlimdv 2666 . . 3  |-  ( A  e.  om  ->  ( E. x  e.  om  B  ~~  x  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) ) )
1110imp 418 . 2  |-  ( ( A  e.  om  /\  E. x  e.  om  B  ~~  x )  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) )
121, 11sylan2b 461 1  |-  ( ( A  e.  om  /\  B  e.  Fin )  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   E.wrex 2544   class class class wbr 4023   suc csuc 4394   omcom 4656    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862   Fincfn 6863
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867
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