ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fidceq Unicode version

Theorem fidceq 6404
Description: Equality of members of a finite set is decidable. This may be counterintuitive: cannot any two sets be elements of a finite set? Well, to show, for example, that  { B ,  C } is finite would require showing it is equinumerous to  1o or to  2o but to show that you'd need to know  B  =  C or  -.  B  =  C, respectively. (Contributed by Jim Kingdon, 5-Sep-2021.)
Assertion
Ref Expression
fidceq  |-  ( ( A  e.  Fin  /\  B  e.  A  /\  C  e.  A )  -> DECID  B  =  C )

Proof of Theorem fidceq
Dummy variables  f  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isfi 6308 . . . 4  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
21biimpi 118 . . 3  |-  ( A  e.  Fin  ->  E. x  e.  om  A  ~~  x
)
323ad2ant1 960 . 2  |-  ( ( A  e.  Fin  /\  B  e.  A  /\  C  e.  A )  ->  E. x  e.  om  A  ~~  x )
4 bren 6294 . . . . 5  |-  ( A 
~~  x  <->  E. f 
f : A -1-1-onto-> x )
54biimpi 118 . . . 4  |-  ( A 
~~  x  ->  E. f 
f : A -1-1-onto-> x )
65ad2antll 475 . . 3  |-  ( ( ( A  e.  Fin  /\  B  e.  A  /\  C  e.  A )  /\  ( x  e.  om  /\  A  ~~  x ) )  ->  E. f 
f : A -1-1-onto-> x )
7 f1of 5157 . . . . . . . . . 10  |-  ( f : A -1-1-onto-> x  ->  f : A --> x )
87adantl 271 . . . . . . . . 9  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A  /\  C  e.  A
)  /\  ( x  e.  om  /\  A  ~~  x ) )  /\  f : A -1-1-onto-> x )  ->  f : A --> x )
9 simpll2 979 . . . . . . . . 9  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A  /\  C  e.  A
)  /\  ( x  e.  om  /\  A  ~~  x ) )  /\  f : A -1-1-onto-> x )  ->  B  e.  A )
108, 9ffvelrnd 5335 . . . . . . . 8  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A  /\  C  e.  A
)  /\  ( x  e.  om  /\  A  ~~  x ) )  /\  f : A -1-1-onto-> x )  ->  (
f `  B )  e.  x )
11 simplrl 502 . . . . . . . 8  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A  /\  C  e.  A
)  /\  ( x  e.  om  /\  A  ~~  x ) )  /\  f : A -1-1-onto-> x )  ->  x  e.  om )
12 elnn 4354 . . . . . . . 8  |-  ( ( ( f `  B
)  e.  x  /\  x  e.  om )  ->  ( f `  B
)  e.  om )
1310, 11, 12syl2anc 403 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A  /\  C  e.  A
)  /\  ( x  e.  om  /\  A  ~~  x ) )  /\  f : A -1-1-onto-> x )  ->  (
f `  B )  e.  om )
14 simpll3 980 . . . . . . . . 9  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A  /\  C  e.  A
)  /\  ( x  e.  om  /\  A  ~~  x ) )  /\  f : A -1-1-onto-> x )  ->  C  e.  A )
158, 14ffvelrnd 5335 . . . . . . . 8  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A  /\  C  e.  A
)  /\  ( x  e.  om  /\  A  ~~  x ) )  /\  f : A -1-1-onto-> x )  ->  (
f `  C )  e.  x )
16 elnn 4354 . . . . . . . 8  |-  ( ( ( f `  C
)  e.  x  /\  x  e.  om )  ->  ( f `  C
)  e.  om )
1715, 11, 16syl2anc 403 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A  /\  C  e.  A
)  /\  ( x  e.  om  /\  A  ~~  x ) )  /\  f : A -1-1-onto-> x )  ->  (
f `  C )  e.  om )
18 nndceq 6143 . . . . . . 7  |-  ( ( ( f `  B
)  e.  om  /\  ( f `  C
)  e.  om )  -> DECID  ( f `  B )  =  ( f `  C ) )
1913, 17, 18syl2anc 403 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A  /\  C  e.  A
)  /\  ( x  e.  om  /\  A  ~~  x ) )  /\  f : A -1-1-onto-> x )  -> DECID  ( f `  B
)  =  ( f `
 C ) )
20 exmiddc 778 . . . . . 6  |-  (DECID  ( f `
 B )  =  ( f `  C
)  ->  ( (
f `  B )  =  ( f `  C )  \/  -.  ( f `  B
)  =  ( f `
 C ) ) )
2119, 20syl 14 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A  /\  C  e.  A
)  /\  ( x  e.  om  /\  A  ~~  x ) )  /\  f : A -1-1-onto-> x )  ->  (
( f `  B
)  =  ( f `
 C )  \/ 
-.  ( f `  B )  =  ( f `  C ) ) )
22 f1of1 5156 . . . . . . . 8  |-  ( f : A -1-1-onto-> x  ->  f : A -1-1-> x )
2322adantl 271 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A  /\  C  e.  A
)  /\  ( x  e.  om  /\  A  ~~  x ) )  /\  f : A -1-1-onto-> x )  ->  f : A -1-1-> x )
24 f1veqaeq 5440 . . . . . . 7  |-  ( ( f : A -1-1-> x  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( f `  B
)  =  ( f `
 C )  ->  B  =  C )
)
2523, 9, 14, 24syl12anc 1168 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A  /\  C  e.  A
)  /\  ( x  e.  om  /\  A  ~~  x ) )  /\  f : A -1-1-onto-> x )  ->  (
( f `  B
)  =  ( f `
 C )  ->  B  =  C )
)
26 fveq2 5209 . . . . . . . 8  |-  ( B  =  C  ->  (
f `  B )  =  ( f `  C ) )
2726con3i 595 . . . . . . 7  |-  ( -.  ( f `  B
)  =  ( f `
 C )  ->  -.  B  =  C
)
2827a1i 9 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A  /\  C  e.  A
)  /\  ( x  e.  om  /\  A  ~~  x ) )  /\  f : A -1-1-onto-> x )  ->  ( -.  ( f `  B
)  =  ( f `
 C )  ->  -.  B  =  C
) )
2925, 28orim12d 733 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A  /\  C  e.  A
)  /\  ( x  e.  om  /\  A  ~~  x ) )  /\  f : A -1-1-onto-> x )  ->  (
( ( f `  B )  =  ( f `  C )  \/  -.  ( f `
 B )  =  ( f `  C
) )  ->  ( B  =  C  \/  -.  B  =  C
) ) )
3021, 29mpd 13 . . . 4  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A  /\  C  e.  A
)  /\  ( x  e.  om  /\  A  ~~  x ) )  /\  f : A -1-1-onto-> x )  ->  ( B  =  C  \/  -.  B  =  C
) )
31 df-dc 777 . . . 4  |-  (DECID  B  =  C  <->  ( B  =  C  \/  -.  B  =  C ) )
3230, 31sylibr 132 . . 3  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A  /\  C  e.  A
)  /\  ( x  e.  om  /\  A  ~~  x ) )  /\  f : A -1-1-onto-> x )  -> DECID  B  =  C
)
336, 32exlimddv 1820 . 2  |-  ( ( ( A  e.  Fin  /\  B  e.  A  /\  C  e.  A )  /\  ( x  e.  om  /\  A  ~~  x ) )  -> DECID  B  =  C
)
343, 33rexlimddv 2482 1  |-  ( ( A  e.  Fin  /\  B  e.  A  /\  C  e.  A )  -> DECID  B  =  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 662  DECID wdc 776    /\ w3a 920    = wceq 1285   E.wex 1422    e. wcel 1434   E.wrex 2350   class class class wbr 3793   omcom 4339   -->wf 4928   -1-1->wf1 4929   -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-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  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-tr 3884  df-id 4056  df-iord 4129  df-on 4131  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-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-en 6288  df-fin 6290
This theorem is referenced by:  fidifsnen  6405  fidifsnid  6406  unfiexmid  6438
  Copyright terms: Public domain W3C validator