Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 6264 | . . . 4 | |
2 | 1 | biimpi 118 | . . 3 |
3 | 2 | 3ad2ant1 959 | . 2 |
4 | bren 6251 | . . . . 5 | |
5 | 4 | biimpi 118 | . . . 4 |
6 | 5 | ad2antll 474 | . . 3 |
7 | f1of 5146 | . . . . . . . . . 10 | |
8 | 7 | adantl 271 | . . . . . . . . 9 |
9 | simpll2 978 | . . . . . . . . 9 | |
10 | 8, 9 | ffvelrnd 5324 | . . . . . . . 8 |
11 | simplrl 501 | . . . . . . . 8 | |
12 | elnn 4346 | . . . . . . . 8 | |
13 | 10, 11, 12 | syl2anc 403 | . . . . . . 7 |
14 | simpll3 979 | . . . . . . . . 9 | |
15 | 8, 14 | ffvelrnd 5324 | . . . . . . . 8 |
16 | elnn 4346 | . . . . . . . 8 | |
17 | 15, 11, 16 | syl2anc 403 | . . . . . . 7 |
18 | nndceq 6100 | . . . . . . 7 DECID | |
19 | 13, 17, 18 | syl2anc 403 | . . . . . 6 DECID |
20 | exmiddc 777 | . . . . . 6 DECID | |
21 | 19, 20 | syl 14 | . . . . 5 |
22 | f1of1 5145 | . . . . . . . 8 | |
23 | 22 | adantl 271 | . . . . . . 7 |
24 | f1veqaeq 5429 | . . . . . . 7 | |
25 | 23, 9, 14, 24 | syl12anc 1167 | . . . . . 6 |
26 | fveq2 5198 | . . . . . . . 8 | |
27 | 26 | con3i 594 | . . . . . . 7 |
28 | 27 | a1i 9 | . . . . . 6 |
29 | 25, 28 | orim12d 732 | . . . . 5 |
30 | 21, 29 | mpd 13 | . . . 4 |
31 | df-dc 776 | . . . 4 DECID | |
32 | 30, 31 | sylibr 132 | . . 3 DECID |
33 | 6, 32 | exlimddv 1819 | . 2 DECID |
34 | 3, 33 | rexlimddv 2481 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wo 661 DECID wdc 775 w3a 919 wceq 1284 wex 1421 wcel 1433 wrex 2349 class class class wbr 3785 com 4331 wf 4918 wf1 4919 wf1o 4921 cfv 4922 cen 6242 cfn 6244 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-en 6245 df-fin 6247 |
This theorem is referenced by: fidifsnen 6355 fidifsnid 6356 |
Copyright terms: Public domain | W3C validator |