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Theorem ficard 8404
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 7098 . . 3  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
2 carden 8390 . . . . 5  |-  ( ( A  e.  V  /\  x  e.  om )  ->  ( ( card `  A
)  =  ( card `  x )  <->  A  ~~  x ) )
3 cardnn 7814 . . . . . . . 8  |-  ( x  e.  om  ->  ( card `  x )  =  x )
4 eqtr 2429 . . . . . . . . 9  |-  ( ( ( card `  A
)  =  ( card `  x )  /\  ( card `  x )  =  x )  ->  ( card `  A )  =  x )
54expcom 425 . . . . . . . 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 2481 . . . . . . 7  |-  ( x  e.  om  ->  (
( card `  A )  =  x  ->  ( card `  A )  e.  om ) )
86, 7syld 42 . . . . . 6  |-  ( x  e.  om  ->  (
( card `  A )  =  ( card `  x
)  ->  ( card `  A )  e.  om ) )
98adantl 453 . . . . 5  |-  ( ( A  e.  V  /\  x  e.  om )  ->  ( ( card `  A
)  =  ( card `  x )  ->  ( card `  A )  e. 
om ) )
102, 9sylbird 227 . . . 4  |-  ( ( A  e.  V  /\  x  e.  om )  ->  ( A  ~~  x  ->  ( card `  A
)  e.  om )
)
1110rexlimdva 2798 . . 3  |-  ( A  e.  V  ->  ( E. x  e.  om  A  ~~  x  ->  ( card `  A )  e. 
om ) )
121, 11syl5bi 209 . 2  |-  ( A  e.  V  ->  ( A  e.  Fin  ->  ( card `  A )  e. 
om ) )
13 cardnn 7814 . . . . . . . 8  |-  ( (
card `  A )  e.  om  ->  ( card `  ( card `  A
) )  =  (
card `  A )
)
1413eqcomd 2417 . . . . . . 7  |-  ( (
card `  A )  e.  om  ->  ( card `  A )  =  (
card `  ( card `  A ) ) )
1514adantl 453 . . . . . 6  |-  ( ( A  e.  V  /\  ( card `  A )  e.  om )  ->  ( card `  A )  =  ( card `  ( card `  A ) ) )
16 carden 8390 . . . . . 6  |-  ( ( A  e.  V  /\  ( card `  A )  e.  om )  ->  (
( card `  A )  =  ( card `  ( card `  A ) )  <-> 
A  ~~  ( card `  A ) ) )
1715, 16mpbid 202 . . . . 5  |-  ( ( A  e.  V  /\  ( card `  A )  e.  om )  ->  A  ~~  ( card `  A
) )
1817ex 424 . . . 4  |-  ( A  e.  V  ->  (
( card `  A )  e.  om  ->  A  ~~  ( card `  A )
) )
1918ancld 537 . . 3  |-  ( A  e.  V  ->  (
( card `  A )  e.  om  ->  ( ( card `  A )  e. 
om  /\  A  ~~  ( card `  A )
) ) )
20 breq2 4184 . . . . 5  |-  ( x  =  ( card `  A
)  ->  ( A  ~~  x  <->  A  ~~  ( card `  A ) ) )
2120rspcev 3020 . . . 4  |-  ( ( ( card `  A
)  e.  om  /\  A  ~~  ( card `  A
) )  ->  E. x  e.  om  A  ~~  x
)
2221, 1sylibr 204 . . 3  |-  ( ( ( card `  A
)  e.  om  /\  A  ~~  ( card `  A
) )  ->  A  e.  Fin )
2319, 22syl6 31 . 2  |-  ( A  e.  V  ->  (
( card `  A )  e.  om  ->  A  e.  Fin ) )
2412, 23impbid 184 1  |-  ( A  e.  V  ->  ( A  e.  Fin  <->  ( card `  A )  e.  om ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2675   class class class wbr 4180   omcom 4812   ` cfv 5421    ~~ cen 7073   Fincfn 7076   cardccrd 7786
This theorem is referenced by:  cfpwsdom  8423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-ac2 8307
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6516  df-recs 6600  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-card 7790  df-ac 7961
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