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Theorem fin0or 6864
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 6739 . . 3  |-  ( A  e.  Fin  <->  E. n  e.  om  A  ~~  n
)
21biimpi 119 . 2  |-  ( A  e.  Fin  ->  E. n  e.  om  A  ~~  n
)
3 nn0suc 4588 . . . 4  |-  ( n  e.  om  ->  (
n  =  (/)  \/  E. m  e.  om  n  =  suc  m ) )
43ad2antrl 487 . . 3  |-  ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  ( n  =  (/)  \/  E. m  e.  om  n  =  suc  m ) )
5 simplrr 531 . . . . . . 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 4015 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  ~~  (/) )
8 en0 6773 . . . . . 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 531 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e. 
om )  ->  A  ~~  n )
1211adantr 274 . . . . . . . . 9  |-  ( ( ( ( A  e. 
Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  ->  A  ~~  n )
1312ensymd 6761 . . . . . . . 8  |-  ( ( ( ( A  e. 
Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  m  e.  om )  /\  n  =  suc  m )  ->  n  ~~  A )
14 bren 6725 . . . . . . . 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 5442 . . . . . . . . . 10  |-  ( f : n -1-1-onto-> A  ->  f :
n --> A )
1716adantl 275 . . . . . . . . 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 4401 . . . . . . . . . . 11  |-  ( m  e.  om  ->  m  e.  suc  m )
1918ad3antlr 490 . . . . . . . . . 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 525 . . . . . . . . . 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 2250 . . . . . . . . 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 5632 . . . . . . . 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 2746 . . . . . . . 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 1891 . . . . . 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 2587 . . . 4  |-  ( ( A  e.  Fin  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  ( E. m  e.  om  n  =  suc  m  ->  E. x  x  e.  A )
)
2810, 27orim12d 781 . . 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 2592 1  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  E. x  x  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 703    = wceq 1348   E.wex 1485    e. wcel 2141   E.wrex 2449   (/)c0 3414   class class class wbr 3989   suc csuc 4350   omcom 4574   -->wf 5194   -1-1-onto->wf1o 5197   ` cfv 5198    ~~ cen 6716   Fincfn 6718
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-er 6513  df-en 6719  df-fin 6721
This theorem is referenced by:  xpfi  6907  fival  6947  fiubm  10763  fsumcllem  11362  fprodcllem  11569
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