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Mirrors > Home > ILE Home > Th. List > dcfi | Unicode version |
Description: Decidability of a family of propositions indexed by a finite set. (Contributed by Jim Kingdon, 30-Sep-2024.) |
Ref | Expression |
---|---|
dcfi | DECID DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleq 2659 | . . 3 | |
2 | 1 | dcbid 828 | . 2 DECID DECID |
3 | raleq 2659 | . . 3 | |
4 | 3 | dcbid 828 | . 2 DECID DECID |
5 | raleq 2659 | . . 3 | |
6 | 5 | dcbid 828 | . 2 DECID DECID |
7 | raleq 2659 | . . 3 | |
8 | 7 | dcbid 828 | . 2 DECID DECID |
9 | ral0 3505 | . . . . 5 | |
10 | 9 | orci 721 | . . . 4 |
11 | df-dc 825 | . . . 4 DECID | |
12 | 10, 11 | mpbir 145 | . . 3 DECID |
13 | 12 | a1i 9 | . 2 DECID DECID |
14 | simpr 109 | . . . . 5 DECID DECID DECID | |
15 | simplrr 526 | . . . . . . . 8 DECID DECID | |
16 | 15 | eldifad 3122 | . . . . . . 7 DECID DECID |
17 | simp-4r 532 | . . . . . . 7 DECID DECID DECID | |
18 | nfsbc1v 2964 | . . . . . . . . 9 | |
19 | 18 | nfdc 1646 | . . . . . . . 8 DECID |
20 | sbceq1a 2955 | . . . . . . . . 9 | |
21 | 20 | dcbid 828 | . . . . . . . 8 DECID DECID |
22 | 19, 21 | rspc 2819 | . . . . . . 7 DECID DECID |
23 | 16, 17, 22 | sylc 62 | . . . . . 6 DECID DECID DECID |
24 | ralsnsg 3607 | . . . . . . . 8 | |
25 | 24 | elv 2725 | . . . . . . 7 |
26 | 25 | dcbii 830 | . . . . . 6 DECID DECID |
27 | 23, 26 | sylibr 133 | . . . . 5 DECID DECID DECID |
28 | dcan 923 | . . . . 5 DECID DECID DECID | |
29 | 14, 27, 28 | sylc 62 | . . . 4 DECID DECID DECID |
30 | ralunb 3298 | . . . . 5 | |
31 | 30 | dcbii 830 | . . . 4 DECID DECID |
32 | 29, 31 | sylibr 133 | . . 3 DECID DECID DECID |
33 | 32 | ex 114 | . 2 DECID DECID DECID |
34 | simpl 108 | . 2 DECID | |
35 | 2, 4, 6, 8, 13, 33, 34 | findcard2sd 6849 | 1 DECID DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 DECID wdc 824 wceq 1342 wcel 2135 wral 2442 cvv 2721 wsbc 2946 cdif 3108 cun 3109 wss 3111 c0 3404 csn 3570 cfn 6697 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-er 6492 df-en 6698 df-fin 6700 |
This theorem is referenced by: prmdc 12041 |
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