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Theorem fidceq 6639
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 6532 . . . 4  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
21biimpi 119 . . 3  |-  ( A  e.  Fin  ->  E. x  e.  om  A  ~~  x
)
323ad2ant1 965 . 2  |-  ( ( A  e.  Fin  /\  B  e.  A  /\  C  e.  A )  ->  E. x  e.  om  A  ~~  x )
4 bren 6518 . . . . 5  |-  ( A 
~~  x  <->  E. f 
f : A -1-1-onto-> x )
54biimpi 119 . . . 4  |-  ( A 
~~  x  ->  E. f 
f : A -1-1-onto-> x )
65ad2antll 476 . . 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 5266 . . . . . . . . . 10  |-  ( f : A -1-1-onto-> x  ->  f : A --> x )
87adantl 272 . . . . . . . . 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 984 . . . . . . . . 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 5449 . . . . . . . 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 503 . . . . . . . 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 4433 . . . . . . . 8  |-  ( ( ( f `  B
)  e.  x  /\  x  e.  om )  ->  ( f `  B
)  e.  om )
1310, 11, 12syl2anc 404 . . . . . . 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 985 . . . . . . . . 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 5449 . . . . . . . 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 4433 . . . . . . . 8  |-  ( ( ( f `  C
)  e.  x  /\  x  e.  om )  ->  ( f `  C
)  e.  om )
1715, 11, 16syl2anc 404 . . . . . . 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 6274 . . . . . . 7  |-  ( ( ( f `  B
)  e.  om  /\  ( f `  C
)  e.  om )  -> DECID  ( f `  B )  =  ( f `  C ) )
1913, 17, 18syl2anc 404 . . . . . 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 783 . . . . . 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 5265 . . . . . . . 8  |-  ( f : A -1-1-onto-> x  ->  f : A -1-1-> x )
2322adantl 272 . . . . . . 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 5562 . . . . . . 7  |-  ( ( f : A -1-1-> x  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( f `  B
)  =  ( f `
 C )  ->  B  =  C )
)
2523, 9, 14, 24syl12anc 1173 . . . . . 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 5318 . . . . . . . 8  |-  ( B  =  C  ->  (
f `  B )  =  ( f `  C ) )
2726con3i 598 . . . . . . 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 736 . . . . 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 782 . . . 4  |-  (DECID  B  =  C  <->  ( B  =  C  \/  -.  B  =  C ) )
3230, 31sylibr 133 . . 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 1827 . 2  |-  ( ( ( A  e.  Fin  /\  B  e.  A  /\  C  e.  A )  /\  ( x  e.  om  /\  A  ~~  x ) )  -> DECID  B  =  C
)
343, 33rexlimddv 2494 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 103    \/ wo 665  DECID wdc 781    /\ w3a 925    = wceq 1290   E.wex 1427    e. wcel 1439   E.wrex 2361   class class class wbr 3851   omcom 4418   -->wf 5024   -1-1->wf1 5025   -1-1-onto->wf1o 5027   ` cfv 5028    ~~ cen 6509   Fincfn 6511
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-br 3852  df-opab 3906  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-en 6512  df-fin 6514
This theorem is referenced by:  fidifsnen  6640  fidifsnid  6641  unfiexmid  6682  undiffi  6689  fidcenumlemim  6715
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