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Theorem fisucdomOLD 7314
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 7133 . 2  |-  ( B  e.  Fin  <->  E. x  e.  om  B  ~~  x
)
2 omsucdomOLD 7304 . . . . . . 7  |-  ( ( A  e.  om  /\  x  e.  om )  ->  ( A  ~<  x  <->  suc 
A  ~<_  x ) )
32adantr 453 . . . . . 6  |-  ( ( ( A  e.  om  /\  x  e.  om )  /\  B  ~~  x )  ->  ( A  ~<  x  <->  suc  A  ~<_  x ) )
4 sdomen2 7254 . . . . . . 7  |-  ( B 
~~  x  ->  ( A  ~<  B  <->  A  ~<  x ) )
54adantl 454 . . . . . 6  |-  ( ( ( A  e.  om  /\  x  e.  om )  /\  B  ~~  x )  ->  ( A  ~<  B  <-> 
A  ~<  x ) )
6 domen2 7252 . . . . . . 7  |-  ( B 
~~  x  ->  ( suc  A  ~<_  B  <->  suc  A  ~<_  x ) )
76adantl 454 . . . . . 6  |-  ( ( ( A  e.  om  /\  x  e.  om )  /\  B  ~~  x )  ->  ( suc  A  ~<_  B 
<->  suc  A  ~<_  x ) )
83, 5, 73bitr4d 278 . . . . 5  |-  ( ( ( A  e.  om  /\  x  e.  om )  /\  B  ~~  x )  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) )
98exp31 589 . . . 4  |-  ( A  e.  om  ->  (
x  e.  om  ->  ( B  ~~  x  -> 
( A  ~<  B  <->  suc  A  ~<_  B ) ) ) )
109rexlimdv 2831 . . 3  |-  ( A  e.  om  ->  ( E. x  e.  om  B  ~~  x  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) ) )
1110imp 420 . 2  |-  ( ( A  e.  om  /\  E. x  e.  om  B  ~~  x )  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) )
121, 11sylan2b 463 1  |-  ( ( A  e.  om  /\  B  e.  Fin )  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    e. wcel 1726   E.wrex 2708   class class class wbr 4214   suc csuc 4585   omcom 4847    ~~ cen 7108    ~<_ cdom 7109    ~< csdm 7110   Fincfn 7111
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115
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