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Theorem fisucdomOLD 7742
Description: TODO-NM: which theorem to be used instead? Keep (and rename) this theorem? 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 7558 . 2  |-  ( B  e.  Fin  <->  E. x  e.  om  B  ~~  x
)
2 omsucdomOLD 7732 . . . . . . 7  |-  ( ( A  e.  om  /\  x  e.  om )  ->  ( A  ~<  x  <->  suc 
A  ~<_  x ) )
32adantr 465 . . . . . 6  |-  ( ( ( A  e.  om  /\  x  e.  om )  /\  B  ~~  x )  ->  ( A  ~<  x  <->  suc  A  ~<_  x ) )
4 sdomen2 7681 . . . . . . 7  |-  ( B 
~~  x  ->  ( A  ~<  B  <->  A  ~<  x ) )
54adantl 466 . . . . . 6  |-  ( ( ( A  e.  om  /\  x  e.  om )  /\  B  ~~  x )  ->  ( A  ~<  B  <-> 
A  ~<  x ) )
6 domen2 7679 . . . . . . 7  |-  ( B 
~~  x  ->  ( suc  A  ~<_  B  <->  suc  A  ~<_  x ) )
76adantl 466 . . . . . 6  |-  ( ( ( A  e.  om  /\  x  e.  om )  /\  B  ~~  x )  ->  ( suc  A  ~<_  B 
<->  suc  A  ~<_  x ) )
83, 5, 73bitr4d 285 . . . . 5  |-  ( ( ( A  e.  om  /\  x  e.  om )  /\  B  ~~  x )  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) )
98exp31 604 . . . 4  |-  ( A  e.  om  ->  (
x  e.  om  ->  ( B  ~~  x  -> 
( A  ~<  B  <->  suc  A  ~<_  B ) ) ) )
109rexlimdv 2947 . . 3  |-  ( A  e.  om  ->  ( E. x  e.  om  B  ~~  x  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) ) )
1110imp 429 . 2  |-  ( ( A  e.  om  /\  E. x  e.  om  B  ~~  x )  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) )
121, 11sylan2b 475 1  |-  ( ( A  e.  om  /\  B  e.  Fin )  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1819   E.wrex 2808   class class class wbr 4456   suc csuc 4889   omcom 6699    ~~ cen 7532    ~<_ cdom 7533    ~< csdm 7534   Fincfn 7535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539
This theorem is referenced by: (None)
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