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Mirrors > Home > ILE Home > Th. List > ssfirab | Unicode version |
Description: A subset of a finite set is finite if it is defined by a decidable property. (Contributed by Jim Kingdon, 27-May-2022.) |
Ref | Expression |
---|---|
ssfirab.a | |
ssfirab.dc | DECID |
Ref | Expression |
---|---|
ssfirab |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabeq 2678 | . . 3 | |
2 | 1 | eleq1d 2208 | . 2 |
3 | rabeq 2678 | . . 3 | |
4 | 3 | eleq1d 2208 | . 2 |
5 | rabeq 2678 | . . 3 | |
6 | 5 | eleq1d 2208 | . 2 |
7 | rabeq 2678 | . . 3 | |
8 | 7 | eleq1d 2208 | . 2 |
9 | rab0 3391 | . . . 4 | |
10 | 0fin 6778 | . . . 4 | |
11 | 9, 10 | eqeltri 2212 | . . 3 |
12 | 11 | a1i 9 | . 2 |
13 | rabun2 3355 | . . . . 5 | |
14 | sbsbc 2913 | . . . . . . . . . 10 | |
15 | vex 2689 | . . . . . . . . . . 11 | |
16 | ralsns 3562 | . . . . . . . . . . 11 | |
17 | 15, 16 | ax-mp 5 | . . . . . . . . . 10 |
18 | 14, 17 | bitr4i 186 | . . . . . . . . 9 |
19 | rabid2 2607 | . . . . . . . . 9 | |
20 | 18, 19 | sylbb2 137 | . . . . . . . 8 |
21 | 20 | adantl 275 | . . . . . . 7 |
22 | 21 | uneq2d 3230 | . . . . . 6 |
23 | simplr 519 | . . . . . . 7 | |
24 | 15 | a1i 9 | . . . . . . 7 |
25 | simprr 521 | . . . . . . . . . 10 | |
26 | 25 | ad2antrr 479 | . . . . . . . . 9 |
27 | 26 | eldifbd 3083 | . . . . . . . 8 |
28 | elrabi 2837 | . . . . . . . 8 | |
29 | 27, 28 | nsyl 617 | . . . . . . 7 |
30 | unsnfi 6807 | . . . . . . 7 | |
31 | 23, 24, 29, 30 | syl3anc 1216 | . . . . . 6 |
32 | 22, 31 | eqeltrrd 2217 | . . . . 5 |
33 | 13, 32 | eqeltrid 2226 | . . . 4 |
34 | ralsns 3562 | . . . . . . . . . . . 12 | |
35 | 15, 34 | ax-mp 5 | . . . . . . . . . . 11 |
36 | sbsbc 2913 | . . . . . . . . . . 11 | |
37 | sbn 1925 | . . . . . . . . . . 11 | |
38 | 35, 36, 37 | 3bitr2ri 208 | . . . . . . . . . 10 |
39 | rabeq0 3392 | . . . . . . . . . 10 | |
40 | 38, 39 | sylbb2 137 | . . . . . . . . 9 |
41 | 40 | adantl 275 | . . . . . . . 8 |
42 | 41 | uneq2d 3230 | . . . . . . 7 |
43 | un0 3396 | . . . . . . 7 | |
44 | 42, 43 | syl6eq 2188 | . . . . . 6 |
45 | 13, 44 | syl5eq 2184 | . . . . 5 |
46 | simplr 519 | . . . . 5 | |
47 | 45, 46 | eqeltrd 2216 | . . . 4 |
48 | simplrr 525 | . . . . . . 7 | |
49 | 48 | eldifad 3082 | . . . . . 6 |
50 | ssfirab.dc | . . . . . . 7 DECID | |
51 | 50 | ad3antrrr 483 | . . . . . 6 DECID |
52 | nfs1v 1912 | . . . . . . . 8 | |
53 | 52 | nfdc 1637 | . . . . . . 7 DECID |
54 | sbequ12 1744 | . . . . . . . 8 | |
55 | 54 | dcbid 823 | . . . . . . 7 DECID DECID |
56 | 53, 55 | rspc 2783 | . . . . . 6 DECID DECID |
57 | 49, 51, 56 | sylc 62 | . . . . 5 DECID |
58 | exmiddc 821 | . . . . 5 DECID | |
59 | 57, 58 | syl 14 | . . . 4 |
60 | 33, 47, 59 | mpjaodan 787 | . . 3 |
61 | 60 | ex 114 | . 2 |
62 | ssfirab.a | . 2 | |
63 | 2, 4, 6, 8, 12, 61, 62 | findcard2sd 6786 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 DECID wdc 819 wceq 1331 wcel 1480 wsb 1735 wral 2416 crab 2420 cvv 2686 wsbc 2909 cdif 3068 cun 3069 wss 3071 c0 3363 csn 3527 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-coll 4043 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-fal 1337 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-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 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-res 4551 df-ima 4552 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-1o 6313 df-er 6429 df-en 6635 df-fin 6637 |
This theorem is referenced by: ssfidc 6823 phivalfi 11888 hashdvds 11897 phiprmpw 11898 phimullem 11901 hashgcdeq 11904 |
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