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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 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 6241 . . . 4
21biimpi 113 . . 3
4 bren 6228 . . . . 5
54biimpi 113 . . . 4
7 f1of 5126 . . . . . . . . . 10
87adantl 262 . . . . . . . . 9
9 simpll2 944 . . . . . . . . 9
108, 9ffvelrnd 5303 . . . . . . . 8
11 simplrl 487 . . . . . . . 8
12 elnn 4328 . . . . . . . 8
1310, 11, 12syl2anc 391 . . . . . . 7
14 simpll3 945 . . . . . . . . 9
158, 14ffvelrnd 5303 . . . . . . . 8
16 elnn 4328 . . . . . . . 8
1715, 11, 16syl2anc 391 . . . . . . 7
18 nndceq 6077 . . . . . . 7 DECID
1913, 17, 18syl2anc 391 . . . . . 6 DECID
20 exmiddc 744 . . . . . 6 DECID
2119, 20syl 14 . . . . 5
22 f1of1 5125 . . . . . . . 8
2322adantl 262 . . . . . . 7
24 f1veqaeq 5408 . . . . . . 7
2523, 9, 14, 24syl12anc 1133 . . . . . 6
26 fveq2 5178 . . . . . . . 8
2726con3i 562 . . . . . . 7
2827a1i 9 . . . . . 6
2925, 28orim12d 700 . . . . 5
3021, 29mpd 13 . . . 4
31 df-dc 743 . . . 4 DECID
3230, 31sylibr 137 . . 3 DECID
336, 32exlimddv 1778 . 2 DECID
343, 33rexlimddv 2437 1 DECID
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 97   wo 629  DECID wdc 742   w3a 885   wceq 1243  wex 1381   wcel 1393  wrex 2307   class class class wbr 3764  com 4313  wf 4898  wf1 4899  wf1o 4901  cfv 4902   cen 6219  cfn 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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