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Theorem fisucdomOLD 7620
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 7436 . 2  |-  ( B  e.  Fin  <->  E. x  e.  om  B  ~~  x
)
2 omsucdomOLD 7610 . . . . . . 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 7559 . . . . . . 7  |-  ( B 
~~  x  ->  ( A  ~<  B  <->  A  ~<  x ) )
54adantl 466 . . . . . 6  |-  ( ( ( A  e.  om  /\  x  e.  om )  /\  B  ~~  x )  ->  ( A  ~<  B  <-> 
A  ~<  x ) )
6 domen2 7557 . . . . . . 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 2939 . . 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 1758   E.wrex 2796   class class class wbr 4393   suc csuc 4822   omcom 6579    ~~ cen 7410    ~<_ cdom 7411    ~< csdm 7412   Fincfn 7413
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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
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 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-om 6580  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417
This theorem is referenced by: (None)
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