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Theorem fin0 6419
Description: A nonempty finite set has at least one element. (Contributed by Jim Kingdon, 10-Sep-2021.)
Assertion
Ref Expression
fin0  |-  ( A  e.  Fin  ->  ( A  =/=  (/)  <->  E. x  x  e.  A ) )
Distinct variable group:    x, A

Proof of Theorem fin0
Dummy variables  f  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isfi 6308 . . 3  |-  ( A  e.  Fin  <->  E. n  e.  om  A  ~~  n
)
21biimpi 118 . 2  |-  ( A  e.  Fin  ->  E. n  e.  om  A  ~~  n
)
3 simplrr 503 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  ~~  n )
4 simpr 108 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  n  =  (/) )
53, 4breqtrd 3817 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  ~~  (/) )
6 en0 6342 . . . . . 6  |-  ( A 
~~  (/)  <->  A  =  (/) )
75, 6sylib 120 . . . . 5  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  =  (/) )
8 nner 2250 . . . . 5  |-  ( A  =  (/)  ->  -.  A  =/=  (/) )
97, 8syl 14 . . . 4  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  -.  A  =/=  (/) )
10 n0r 3268 . . . . . 6  |-  ( E. x  x  e.  A  ->  A  =/=  (/) )
1110necon2bi 2301 . . . . 5  |-  ( A  =  (/)  ->  -.  E. x  x  e.  A
)
127, 11syl 14 . . . 4  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  -.  E. x  x  e.  A
)
139, 122falsed 651 . . 3  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  ( A  =/=  (/)  <->  E. x  x  e.  A ) )
14 simplrr 503 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e. 
om )  ->  A  ~~  n )
1514adantr 270 . . . . . . . . 9  |-  ( ( ( ( A  e. 
Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  ->  A  ~~  n )
1615ensymd 6330 . . . . . . . 8  |-  ( ( ( ( A  e. 
Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  ->  n  ~~  A )
17 bren 6294 . . . . . . . 8  |-  ( n 
~~  A  <->  E. f 
f : n -1-1-onto-> A )
1816, 17sylib 120 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  ->  E. f 
f : n -1-1-onto-> A )
19 f1of 5157 . . . . . . . . . . . 12  |-  ( f : n -1-1-onto-> A  ->  f :
n --> A )
2019adantl 271 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  Fin  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  /\  f : n -1-1-onto-> A )  ->  f : n --> A )
21 sucidg 4179 . . . . . . . . . . . . 13  |-  ( m  e.  om  ->  m  e.  suc  m )
2221ad3antlr 477 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  Fin  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  /\  f : n -1-1-onto-> A )  ->  m  e.  suc  m )
23 simplr 497 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  Fin  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  /\  f : n -1-1-onto-> A )  ->  n  =  suc  m )
2422, 23eleqtrrd 2159 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  Fin  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  /\  f : n -1-1-onto-> A )  ->  m  e.  n )
2520, 24ffvelrnd 5335 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  Fin  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  /\  f : n -1-1-onto-> A )  ->  (
f `  m )  e.  A )
26 elex2 2616 . . . . . . . . . 10  |-  ( ( f `  m )  e.  A  ->  E. x  x  e.  A )
2725, 26syl 14 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Fin  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  /\  f : n -1-1-onto-> A )  ->  E. x  x  e.  A )
2827, 10syl 14 . . . . . . . 8  |-  ( ( ( ( ( A  e.  Fin  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  /\  f : n -1-1-onto-> A )  ->  A  =/=  (/) )
2928, 272thd 173 . . . . . . 7  |-  ( ( ( ( ( A  e.  Fin  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  /\  f : n -1-1-onto-> A )  ->  ( A  =/=  (/)  <->  E. x  x  e.  A ) )
3018, 29exlimddv 1820 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  ->  ( A  =/=  (/)  <->  E. x  x  e.  A ) )
3130ex 113 . . . . 5  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e. 
om )  ->  (
n  =  suc  m  ->  ( A  =/=  (/)  <->  E. x  x  e.  A )
) )
3231rexlimdva 2478 . . . 4  |-  ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  ( E. m  e.  om  n  =  suc  m  ->  ( A  =/=  (/)  <->  E. x  x  e.  A ) ) )
3332imp 122 . . 3  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  E. m  e.  om  n  =  suc  m )  ->  ( A  =/=  (/)  <->  E. x  x  e.  A ) )
34 nn0suc 4353 . . . 4  |-  ( n  e.  om  ->  (
n  =  (/)  \/  E. m  e.  om  n  =  suc  m ) )
3534ad2antrl 474 . . 3  |-  ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  ( n  =  (/)  \/  E. m  e.  om  n  =  suc  m ) )
3613, 33, 35mpjaodan 745 . 2  |-  ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  ( A  =/=  (/)  <->  E. x  x  e.  A ) )
372, 36rexlimddv 2482 1  |-  ( A  e.  Fin  ->  ( A  =/=  (/)  <->  E. x  x  e.  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    = wceq 1285   E.wex 1422    e. wcel 1434    =/= wne 2246   E.wrex 2350   (/)c0 3258   class class class wbr 3793   suc csuc 4128   omcom 4339   -->wf 4928   -1-1-onto->wf1o 4931   ` cfv 4932    ~~ cen 6285   Fincfn 6287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-id 4056  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-er 6172  df-en 6288  df-fin 6290
This theorem is referenced by:  findcard2  6423  findcard2s  6424  diffisn  6427
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