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Theorem fin0 6832
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 6708 . . 3  |-  ( A  e.  Fin  <->  E. n  e.  om  A  ~~  n
)
21biimpi 119 . 2  |-  ( A  e.  Fin  ->  E. n  e.  om  A  ~~  n
)
3 simplrr 526 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  ~~  n )
4 simpr 109 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  n  =  (/) )
53, 4breqtrd 3992 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  ~~  (/) )
6 en0 6742 . . . . . 6  |-  ( A 
~~  (/)  <->  A  =  (/) )
75, 6sylib 121 . . . . 5  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  =  (/) )
8 nner 2331 . . . . 5  |-  ( A  =  (/)  ->  -.  A  =/=  (/) )
97, 8syl 14 . . . 4  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  -.  A  =/=  (/) )
10 n0r 3408 . . . . . 6  |-  ( E. x  x  e.  A  ->  A  =/=  (/) )
1110necon2bi 2382 . . . . 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 692 . . 3  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  ( A  =/=  (/)  <->  E. x  x  e.  A ) )
14 simplrr 526 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e. 
om )  ->  A  ~~  n )
1514adantr 274 . . . . . . . . 9  |-  ( ( ( ( A  e. 
Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  ->  A  ~~  n )
1615ensymd 6730 . . . . . . . 8  |-  ( ( ( ( A  e. 
Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  ->  n  ~~  A )
17 bren 6694 . . . . . . . 8  |-  ( n 
~~  A  <->  E. f 
f : n -1-1-onto-> A )
1816, 17sylib 121 . . . . . . 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 5416 . . . . . . . . . . . 12  |-  ( f : n -1-1-onto-> A  ->  f :
n --> A )
2019adantl 275 . . . . . . . . . . 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 4378 . . . . . . . . . . . . 13  |-  ( m  e.  om  ->  m  e.  suc  m )
2221ad3antlr 485 . . . . . . . . . . . 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 520 . . . . . . . . . . . 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 2237 . . . . . . . . . . 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 5605 . . . . . . . . . 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 2728 . . . . . . . . . 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 174 . . . . . . 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 1878 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  ->  ( A  =/=  (/)  <->  E. x  x  e.  A ) )
3130ex 114 . . . . 5  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e. 
om )  ->  (
n  =  suc  m  ->  ( A  =/=  (/)  <->  E. x  x  e.  A )
) )
3231rexlimdva 2574 . . . 4  |-  ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  ( E. m  e.  om  n  =  suc  m  ->  ( A  =/=  (/)  <->  E. x  x  e.  A ) ) )
3332imp 123 . . 3  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  E. m  e.  om  n  =  suc  m )  ->  ( A  =/=  (/)  <->  E. x  x  e.  A ) )
34 nn0suc 4565 . . . 4  |-  ( n  e.  om  ->  (
n  =  (/)  \/  E. m  e.  om  n  =  suc  m ) )
3534ad2antrl 482 . . 3  |-  ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  ( n  =  (/)  \/  E. m  e.  om  n  =  suc  m ) )
3613, 33, 35mpjaodan 788 . 2  |-  ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  ( A  =/=  (/)  <->  E. x  x  e.  A ) )
372, 36rexlimddv 2579 1  |-  ( A  e.  Fin  ->  ( A  =/=  (/)  <->  E. x  x  e.  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698    = wceq 1335   E.wex 1472    e. wcel 2128    =/= wne 2327   E.wrex 2436   (/)c0 3395   class class class wbr 3967   suc csuc 4327   omcom 4551   -->wf 5168   -1-1-onto->wf1o 5171   ` cfv 5172    ~~ cen 6685   Fincfn 6687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-iinf 4549
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-br 3968  df-opab 4028  df-id 4255  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-er 6482  df-en 6688  df-fin 6690
This theorem is referenced by:  findcard2  6836  findcard2s  6837  diffisn  6840  fimax2gtri  6848  elfi2  6918  elfir  6919  fiuni  6924  fifo  6926
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