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Mirrors > Home > ILE Home > Th. List > fidceq | Unicode version |
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.) |
Ref | Expression |
---|---|
fidceq | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 6241 | . . . 4 | |
2 | 1 | biimpi 113 | . . 3 |
3 | 2 | 3ad2ant1 925 | . 2 |
4 | bren 6228 | . . . . 5 | |
5 | 4 | biimpi 113 | . . . 4 |
6 | 5 | ad2antll 460 | . . 3 |
7 | f1of 5126 | . . . . . . . . . 10 | |
8 | 7 | adantl 262 | . . . . . . . . 9 |
9 | simpll2 944 | . . . . . . . . 9 | |
10 | 8, 9 | ffvelrnd 5303 | . . . . . . . 8 |
11 | simplrl 487 | . . . . . . . 8 | |
12 | elnn 4328 | . . . . . . . 8 | |
13 | 10, 11, 12 | syl2anc 391 | . . . . . . 7 |
14 | simpll3 945 | . . . . . . . . 9 | |
15 | 8, 14 | ffvelrnd 5303 | . . . . . . . 8 |
16 | elnn 4328 | . . . . . . . 8 | |
17 | 15, 11, 16 | syl2anc 391 | . . . . . . 7 |
18 | nndceq 6077 | . . . . . . 7 DECID | |
19 | 13, 17, 18 | syl2anc 391 | . . . . . 6 DECID |
20 | exmiddc 744 | . . . . . 6 DECID | |
21 | 19, 20 | syl 14 | . . . . 5 |
22 | f1of1 5125 | . . . . . . . 8 | |
23 | 22 | adantl 262 | . . . . . . 7 |
24 | f1veqaeq 5408 | . . . . . . 7 | |
25 | 23, 9, 14, 24 | syl12anc 1133 | . . . . . 6 |
26 | fveq2 5178 | . . . . . . . 8 | |
27 | 26 | con3i 562 | . . . . . . 7 |
28 | 27 | a1i 9 | . . . . . 6 |
29 | 25, 28 | orim12d 700 | . . . . 5 |
30 | 21, 29 | mpd 13 | . . . 4 |
31 | df-dc 743 | . . . 4 DECID | |
32 | 30, 31 | sylibr 137 | . . 3 DECID |
33 | 6, 32 | exlimddv 1778 | . 2 DECID |
34 | 3, 33 | rexlimddv 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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