MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ficard Structured version   Unicode version

Theorem ficard 8853
Description: A set is finite iff its cardinal is a natural number. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
ficard  |-  ( A  e.  V  ->  ( A  e.  Fin  <->  ( card `  A )  e.  om ) )

Proof of Theorem ficard
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 isfi 7458 . . 3  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
2 carden 8839 . . . . 5  |-  ( ( A  e.  V  /\  x  e.  om )  ->  ( ( card `  A
)  =  ( card `  x )  <->  A  ~~  x ) )
3 cardnn 8257 . . . . . . . 8  |-  ( x  e.  om  ->  ( card `  x )  =  x )
4 eqtr 2408 . . . . . . . . 9  |-  ( ( ( card `  A
)  =  ( card `  x )  /\  ( card `  x )  =  x )  ->  ( card `  A )  =  x )
54expcom 433 . . . . . . . 8  |-  ( (
card `  x )  =  x  ->  ( (
card `  A )  =  ( card `  x
)  ->  ( card `  A )  =  x ) )
63, 5syl 16 . . . . . . 7  |-  ( x  e.  om  ->  (
( card `  A )  =  ( card `  x
)  ->  ( card `  A )  =  x ) )
7 eleq1a 2465 . . . . . . 7  |-  ( x  e.  om  ->  (
( card `  A )  =  x  ->  ( card `  A )  e.  om ) )
86, 7syld 44 . . . . . 6  |-  ( x  e.  om  ->  (
( card `  A )  =  ( card `  x
)  ->  ( card `  A )  e.  om ) )
98adantl 464 . . . . 5  |-  ( ( A  e.  V  /\  x  e.  om )  ->  ( ( card `  A
)  =  ( card `  x )  ->  ( card `  A )  e. 
om ) )
102, 9sylbird 235 . . . 4  |-  ( ( A  e.  V  /\  x  e.  om )  ->  ( A  ~~  x  ->  ( card `  A
)  e.  om )
)
1110rexlimdva 2874 . . 3  |-  ( A  e.  V  ->  ( E. x  e.  om  A  ~~  x  ->  ( card `  A )  e. 
om ) )
121, 11syl5bi 217 . 2  |-  ( A  e.  V  ->  ( A  e.  Fin  ->  ( card `  A )  e. 
om ) )
13 cardnn 8257 . . . . . . . 8  |-  ( (
card `  A )  e.  om  ->  ( card `  ( card `  A
) )  =  (
card `  A )
)
1413eqcomd 2390 . . . . . . 7  |-  ( (
card `  A )  e.  om  ->  ( card `  A )  =  (
card `  ( card `  A ) ) )
1514adantl 464 . . . . . 6  |-  ( ( A  e.  V  /\  ( card `  A )  e.  om )  ->  ( card `  A )  =  ( card `  ( card `  A ) ) )
16 carden 8839 . . . . . 6  |-  ( ( A  e.  V  /\  ( card `  A )  e.  om )  ->  (
( card `  A )  =  ( card `  ( card `  A ) )  <-> 
A  ~~  ( card `  A ) ) )
1715, 16mpbid 210 . . . . 5  |-  ( ( A  e.  V  /\  ( card `  A )  e.  om )  ->  A  ~~  ( card `  A
) )
1817ex 432 . . . 4  |-  ( A  e.  V  ->  (
( card `  A )  e.  om  ->  A  ~~  ( card `  A )
) )
1918ancld 551 . . 3  |-  ( A  e.  V  ->  (
( card `  A )  e.  om  ->  ( ( card `  A )  e. 
om  /\  A  ~~  ( card `  A )
) ) )
20 breq2 4371 . . . . 5  |-  ( x  =  ( card `  A
)  ->  ( A  ~~  x  <->  A  ~~  ( card `  A ) ) )
2120rspcev 3135 . . . 4  |-  ( ( ( card `  A
)  e.  om  /\  A  ~~  ( card `  A
) )  ->  E. x  e.  om  A  ~~  x
)
2221, 1sylibr 212 . . 3  |-  ( ( ( card `  A
)  e.  om  /\  A  ~~  ( card `  A
) )  ->  A  e.  Fin )
2319, 22syl6 33 . 2  |-  ( A  e.  V  ->  (
( card `  A )  e.  om  ->  A  e.  Fin ) )
2412, 23impbid 191 1  |-  ( A  e.  V  ->  ( A  e.  Fin  <->  ( card `  A )  e.  om ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   E.wrex 2733   class class class wbr 4367   ` cfv 5496   omcom 6599    ~~ cen 7432   Fincfn 7435   cardccrd 8229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-ac2 8756
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-om 6600  df-recs 6960  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-ac 8410
This theorem is referenced by:  cfpwsdom  8872
  Copyright terms: Public domain W3C validator