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Theorem fin0or 6746
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 6621 . . 3  |-  ( A  e.  Fin  <->  E. n  e.  om  A  ~~  n
)
21biimpi 119 . 2  |-  ( A  e.  Fin  ->  E. n  e.  om  A  ~~  n
)
3 nn0suc 4486 . . . 4  |-  ( n  e.  om  ->  (
n  =  (/)  \/  E. m  e.  om  n  =  suc  m ) )
43ad2antrl 479 . . 3  |-  ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  ( n  =  (/)  \/  E. m  e.  om  n  =  suc  m ) )
5 simplrr 508 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  ~~  n )
6 simpr 109 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  n  =  (/) )
75, 6breqtrd 3922 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  ~~  (/) )
8 en0 6655 . . . . . 6  |-  ( A 
~~  (/)  <->  A  =  (/) )
97, 8sylib 121 . . . . 5  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  =  (/) )
109ex 114 . . . 4  |-  ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  ( n  =  (/)  ->  A  =  (/) ) )
11 simplrr 508 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e. 
om )  ->  A  ~~  n )
1211adantr 272 . . . . . . . . 9  |-  ( ( ( ( A  e. 
Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  ->  A  ~~  n )
1312ensymd 6643 . . . . . . . 8  |-  ( ( ( ( A  e. 
Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  ->  n  ~~  A )
14 bren 6607 . . . . . . . 8  |-  ( n 
~~  A  <->  E. f 
f : n -1-1-onto-> A )
1513, 14sylib 121 . . . . . . 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 5333 . . . . . . . . . 10  |-  ( f : n -1-1-onto-> A  ->  f :
n --> A )
1716adantl 273 . . . . . . . . 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 4306 . . . . . . . . . . 11  |-  ( m  e.  om  ->  m  e.  suc  m )
1918ad3antlr 482 . . . . . . . . . 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 502 . . . . . . . . . 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 2195 . . . . . . . . 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, 21ffvelrnd 5522 . . . . . . . 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 2674 . . . . . . . 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 1852 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  ->  E. x  x  e.  A )
2625ex 114 . . . . 5  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e. 
om )  ->  (
n  =  suc  m  ->  E. x  x  e.  A ) )
2726rexlimdva 2524 . . . 4  |-  ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  ( E. m  e.  om  n  =  suc  m  ->  E. x  x  e.  A )
)
2810, 27orim12d 758 . . 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 2529 1  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  E. x  x  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 680    = wceq 1314   E.wex 1451    e. wcel 1463   E.wrex 2392   (/)c0 3331   class class class wbr 3897   suc csuc 4255   omcom 4472   -->wf 5087   -1-1-onto->wf1o 5090   ` cfv 5091    ~~ cen 6598   Fincfn 6600
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-id 4183  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-er 6395  df-en 6601  df-fin 6603
This theorem is referenced by:  xpfi  6784  fival  6824  fsumcllem  11108
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