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Theorem fidceq 6330
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 6241 . . . 4  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
21biimpi 113 . . 3  |-  ( A  e.  Fin  ->  E. x  e.  om  A  ~~  x
)
323ad2ant1 925 . 2  |-  ( ( A  e.  Fin  /\  B  e.  A  /\  C  e.  A )  ->  E. x  e.  om  A  ~~  x )
4 bren 6228 . . . . 5  |-  ( A 
~~  x  <->  E. f 
f : A -1-1-onto-> x )
54biimpi 113 . . . 4  |-  ( A 
~~  x  ->  E. f 
f : A -1-1-onto-> x )
65ad2antll 460 . . 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 5126 . . . . . . . . . 10  |-  ( f : A -1-1-onto-> x  ->  f : A --> x )
87adantl 262 . . . . . . . . 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 944 . . . . . . . . 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 5303 . . . . . . . 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 487 . . . . . . . 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 4328 . . . . . . . 8  |-  ( ( ( f `  B
)  e.  x  /\  x  e.  om )  ->  ( f `  B
)  e.  om )
1310, 11, 12syl2anc 391 . . . . . . 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 945 . . . . . . . . 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 5303 . . . . . . . 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 4328 . . . . . . . 8  |-  ( ( ( f `  C
)  e.  x  /\  x  e.  om )  ->  ( f `  C
)  e.  om )
1715, 11, 16syl2anc 391 . . . . . . 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 6077 . . . . . . 7  |-  ( ( ( f `  B
)  e.  om  /\  ( f `  C
)  e.  om )  -> DECID  ( f `  B )  =  ( f `  C ) )
1913, 17, 18syl2anc 391 . . . . . 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 744 . . . . . 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 5125 . . . . . . . 8  |-  ( f : A -1-1-onto-> x  ->  f : A -1-1-> x )
2322adantl 262 . . . . . . 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 5408 . . . . . . 7  |-  ( ( f : A -1-1-> x  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( f `  B
)  =  ( f `
 C )  ->  B  =  C )
)
2523, 9, 14, 24syl12anc 1133 . . . . . 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 5178 . . . . . . . 8  |-  ( B  =  C  ->  (
f `  B )  =  ( f `  C ) )
2726con3i 562 . . . . . . 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 700 . . . . 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 743 . . . 4  |-  (DECID  B  =  C  <->  ( B  =  C  \/  -.  B  =  C ) )
3230, 31sylibr 137 . . 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 1778 . 2  |-  ( ( ( A  e.  Fin  /\  B  e.  A  /\  C  e.  A )  /\  ( x  e.  om  /\  A  ~~  x ) )  -> DECID  B  =  C
)
343, 33rexlimddv 2437 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 97    \/ wo 629  DECID wdc 742    /\ w3a 885    = wceq 1243   E.wex 1381    e. wcel 1393   E.wrex 2307   class class class wbr 3764   omcom 4313   -->wf 4898   -1-1->wf1 4899   -1-1-onto->wf1o 4901   ` cfv 4902    ~~ cen 6219   Fincfn 6221
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-opab 3819  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-en 6222  df-fin 6224
This theorem is referenced by:  fidifsnen  6331  fidifsnid  6332
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