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Theorem fidceq 6886
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 6778 . . . 4  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
21biimpi 120 . . 3  |-  ( A  e.  Fin  ->  E. x  e.  om  A  ~~  x
)
323ad2ant1 1019 . 2  |-  ( ( A  e.  Fin  /\  B  e.  A  /\  C  e.  A )  ->  E. x  e.  om  A  ~~  x )
4 bren 6764 . . . . 5  |-  ( A 
~~  x  <->  E. f 
f : A -1-1-onto-> x )
54biimpi 120 . . . 4  |-  ( A 
~~  x  ->  E. f 
f : A -1-1-onto-> x )
65ad2antll 491 . . 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 5475 . . . . . . . . . 10  |-  ( f : A -1-1-onto-> x  ->  f : A --> x )
87adantl 277 . . . . . . . . 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 1038 . . . . . . . . 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, 9ffvelcdmd 5667 . . . . . . . 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 535 . . . . . . . 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 4619 . . . . . . . 8  |-  ( ( ( f `  B
)  e.  x  /\  x  e.  om )  ->  ( f `  B
)  e.  om )
1310, 11, 12syl2anc 411 . . . . . . 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 1039 . . . . . . . . 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, 14ffvelcdmd 5667 . . . . . . . 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 4619 . . . . . . . 8  |-  ( ( ( f `  C
)  e.  x  /\  x  e.  om )  ->  ( f `  C
)  e.  om )
1715, 11, 16syl2anc 411 . . . . . . 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 6517 . . . . . . 7  |-  ( ( ( f `  B
)  e.  om  /\  ( f `  C
)  e.  om )  -> DECID  ( f `  B )  =  ( f `  C ) )
1913, 17, 18syl2anc 411 . . . . . 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 837 . . . . . 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 5474 . . . . . . . 8  |-  ( f : A -1-1-onto-> x  ->  f : A -1-1-> x )
2322adantl 277 . . . . . . 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 5785 . . . . . . 7  |-  ( ( f : A -1-1-> x  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( f `  B
)  =  ( f `
 C )  ->  B  =  C )
)
2523, 9, 14, 24syl12anc 1246 . . . . . 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 5529 . . . . . . . 8  |-  ( B  =  C  ->  (
f `  B )  =  ( f `  C ) )
2726con3i 633 . . . . . . 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 787 . . . . 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 836 . . . 4  |-  (DECID  B  =  C  <->  ( B  =  C  \/  -.  B  =  C ) )
3230, 31sylibr 134 . . 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 1909 . 2  |-  ( ( ( A  e.  Fin  /\  B  e.  A  /\  C  e.  A )  /\  ( x  e.  om  /\  A  ~~  x ) )  -> DECID  B  =  C
)
343, 33rexlimddv 2611 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 104    \/ wo 709  DECID wdc 835    /\ w3a 979    = wceq 1363   E.wex 1502    e. wcel 2159   E.wrex 2468   class class class wbr 4017   omcom 4603   -->wf 5226   -1-1->wf1 5227   -1-1-onto->wf1o 5229   ` cfv 5230    ~~ cen 6755   Fincfn 6757
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-ral 2472  df-rex 2473  df-v 2753  df-sbc 2977  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-br 4018  df-opab 4079  df-tr 4116  df-id 4307  df-iord 4380  df-on 4382  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-en 6758  df-fin 6760
This theorem is referenced by:  fidifsnen  6887  fidifsnid  6888  pw1fin  6927  unfiexmid  6934  undiffi  6941  fidcenumlemim  6968
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