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Theorem fidceq 6763
 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 is finite would require showing it is equinumerous to or to but to show that you'd need to know or , respectively. (Contributed by Jim Kingdon, 5-Sep-2021.)
Assertion
Ref Expression
fidceq DECID

Proof of Theorem fidceq
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isfi 6655 . . . 4
21biimpi 119 . . 3
4 bren 6641 . . . . 5
54biimpi 119 . . . 4
7 f1of 5367 . . . . . . . . . 10
87adantl 275 . . . . . . . . 9
9 simpll2 1021 . . . . . . . . 9
108, 9ffvelrnd 5556 . . . . . . . 8
11 simplrl 524 . . . . . . . 8
12 elnn 4519 . . . . . . . 8
1310, 11, 12syl2anc 408 . . . . . . 7
14 simpll3 1022 . . . . . . . . 9
158, 14ffvelrnd 5556 . . . . . . . 8
16 elnn 4519 . . . . . . . 8
1715, 11, 16syl2anc 408 . . . . . . 7
18 nndceq 6395 . . . . . . 7 DECID
1913, 17, 18syl2anc 408 . . . . . 6 DECID
20 exmiddc 821 . . . . . 6 DECID
2119, 20syl 14 . . . . 5
22 f1of1 5366 . . . . . . . 8
2322adantl 275 . . . . . . 7
24 f1veqaeq 5670 . . . . . . 7
2523, 9, 14, 24syl12anc 1214 . . . . . 6
26 fveq2 5421 . . . . . . . 8
2726con3i 621 . . . . . . 7
2827a1i 9 . . . . . 6
2925, 28orim12d 775 . . . . 5
3021, 29mpd 13 . . . 4
31 df-dc 820 . . . 4 DECID
3230, 31sylibr 133 . . 3 DECID
336, 32exlimddv 1870 . 2 DECID
343, 33rexlimddv 2554 1 DECID
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wo 697  DECID wdc 819   w3a 962   wceq 1331  wex 1468   wcel 1480  wrex 2417   class class class wbr 3929  com 4504  wf 5119  wf1 5120  wf1o 5122  cfv 5123   cen 6632  cfn 6634 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502 This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-en 6635  df-fin 6637 This theorem is referenced by:  fidifsnen  6764  fidifsnid  6765  unfiexmid  6806  undiffi  6813  fidcenumlemim  6840
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