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Theorem fin0or 6885
Description: A finite set is either empty or inhabited. (Contributed by Jim Kingdon, 30-Sep-2021.)
Assertion
Ref Expression
fin0or  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  E. x  x  e.  A
) )
Distinct variable group:    x, A

Proof of Theorem fin0or
Dummy variables  f  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isfi 6760 . . 3  |-  ( A  e.  Fin  <->  E. n  e.  om  A  ~~  n
)
21biimpi 120 . 2  |-  ( A  e.  Fin  ->  E. n  e.  om  A  ~~  n
)
3 nn0suc 4603 . . . 4  |-  ( n  e.  om  ->  (
n  =  (/)  \/  E. m  e.  om  n  =  suc  m ) )
43ad2antrl 490 . . 3  |-  ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  ( n  =  (/)  \/  E. m  e.  om  n  =  suc  m ) )
5 simplrr 536 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  ~~  n )
6 simpr 110 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  n  =  (/) )
75, 6breqtrd 4029 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  ~~  (/) )
8 en0 6794 . . . . . 6  |-  ( A 
~~  (/)  <->  A  =  (/) )
97, 8sylib 122 . . . . 5  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  =  (/) )
109ex 115 . . . 4  |-  ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  ( n  =  (/)  ->  A  =  (/) ) )
11 simplrr 536 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e. 
om )  ->  A  ~~  n )
1211adantr 276 . . . . . . . . 9  |-  ( ( ( ( A  e. 
Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  ->  A  ~~  n )
1312ensymd 6782 . . . . . . . 8  |-  ( ( ( ( A  e. 
Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  ->  n  ~~  A )
14 bren 6746 . . . . . . . 8  |-  ( n 
~~  A  <->  E. f 
f : n -1-1-onto-> A )
1513, 14sylib 122 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  ->  E. f 
f : n -1-1-onto-> A )
16 f1of 5461 . . . . . . . . . 10  |-  ( f : n -1-1-onto-> A  ->  f :
n --> A )
1716adantl 277 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Fin  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  /\  f : n -1-1-onto-> A )  ->  f : n --> A )
18 sucidg 4416 . . . . . . . . . . 11  |-  ( m  e.  om  ->  m  e.  suc  m )
1918ad3antlr 493 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  Fin  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  /\  f : n -1-1-onto-> A )  ->  m  e.  suc  m )
20 simplr 528 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  Fin  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  /\  f : n -1-1-onto-> A )  ->  n  =  suc  m )
2119, 20eleqtrrd 2257 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Fin  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  /\  f : n -1-1-onto-> A )  ->  m  e.  n )
2217, 21ffvelcdmd 5652 . . . . . . . 8  |-  ( ( ( ( ( A  e.  Fin  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  /\  f : n -1-1-onto-> A )  ->  (
f `  m )  e.  A )
23 elex2 2753 . . . . . . . 8  |-  ( ( f `  m )  e.  A  ->  E. x  x  e.  A )
2422, 23syl 14 . . . . . . 7  |-  ( ( ( ( ( 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 )
2515, 24exlimddv 1898 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  ->  E. x  x  e.  A )
2625ex 115 . . . . 5  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e. 
om )  ->  (
n  =  suc  m  ->  E. x  x  e.  A ) )
2726rexlimdva 2594 . . . 4  |-  ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  ( E. m  e.  om  n  =  suc  m  ->  E. x  x  e.  A )
)
2810, 27orim12d 786 . . 3  |-  ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  ( (
n  =  (/)  \/  E. m  e.  om  n  =  suc  m )  -> 
( A  =  (/)  \/ 
E. x  x  e.  A ) ) )
294, 28mpd 13 . 2  |-  ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  ( A  =  (/)  \/  E. x  x  e.  A )
)
302, 29rexlimddv 2599 1  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  E. x  x  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    \/ wo 708    = wceq 1353   E.wex 1492    e. wcel 2148   E.wrex 2456   (/)c0 3422   class class class wbr 4003   suc csuc 4365   omcom 4589   -->wf 5212   -1-1-onto->wf1o 5215   ` cfv 5216    ~~ cen 6737   Fincfn 6739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-id 4293  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-er 6534  df-en 6740  df-fin 6742
This theorem is referenced by:  xpfi  6928  fival  6968  fiubm  10807  fsumcllem  11406  fprodcllem  11613
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