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Mirrors > Home > ILE Home > Th. List > findcard | Unicode version |
Description: Schema for induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on the given set with any one element removed. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
findcard.1 | |
findcard.2 | |
findcard.3 | |
findcard.4 | |
findcard.5 | |
findcard.6 |
Ref | Expression |
---|---|
findcard |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | findcard.4 | . 2 | |
2 | isfi 6655 | . . 3 | |
3 | breq2 3933 | . . . . . . . 8 | |
4 | 3 | imbi1d 230 | . . . . . . 7 |
5 | 4 | albidv 1796 | . . . . . 6 |
6 | breq2 3933 | . . . . . . . 8 | |
7 | 6 | imbi1d 230 | . . . . . . 7 |
8 | 7 | albidv 1796 | . . . . . 6 |
9 | breq2 3933 | . . . . . . . 8 | |
10 | 9 | imbi1d 230 | . . . . . . 7 |
11 | 10 | albidv 1796 | . . . . . 6 |
12 | en0 6689 | . . . . . . . 8 | |
13 | findcard.5 | . . . . . . . . 9 | |
14 | findcard.1 | . . . . . . . . 9 | |
15 | 13, 14 | mpbiri 167 | . . . . . . . 8 |
16 | 12, 15 | sylbi 120 | . . . . . . 7 |
17 | 16 | ax-gen 1425 | . . . . . 6 |
18 | peano2 4509 | . . . . . . . . . . . . 13 | |
19 | breq2 3933 | . . . . . . . . . . . . . 14 | |
20 | 19 | rspcev 2789 | . . . . . . . . . . . . 13 |
21 | 18, 20 | sylan 281 | . . . . . . . . . . . 12 |
22 | isfi 6655 | . . . . . . . . . . . 12 | |
23 | 21, 22 | sylibr 133 | . . . . . . . . . . 11 |
24 | 23 | 3adant2 1000 | . . . . . . . . . 10 |
25 | dif1en 6773 | . . . . . . . . . . . . . . . 16 | |
26 | 25 | 3expa 1181 | . . . . . . . . . . . . . . 15 |
27 | vex 2689 | . . . . . . . . . . . . . . . . 17 | |
28 | difexg 4069 | . . . . . . . . . . . . . . . . 17 | |
29 | 27, 28 | ax-mp 5 | . . . . . . . . . . . . . . . 16 |
30 | breq1 3932 | . . . . . . . . . . . . . . . . 17 | |
31 | findcard.2 | . . . . . . . . . . . . . . . . 17 | |
32 | 30, 31 | imbi12d 233 | . . . . . . . . . . . . . . . 16 |
33 | 29, 32 | spcv 2779 | . . . . . . . . . . . . . . 15 |
34 | 26, 33 | syl5com 29 | . . . . . . . . . . . . . 14 |
35 | 34 | ralrimdva 2512 | . . . . . . . . . . . . 13 |
36 | 35 | imp 123 | . . . . . . . . . . . 12 |
37 | 36 | an32s 557 | . . . . . . . . . . 11 |
38 | 37 | 3impa 1176 | . . . . . . . . . 10 |
39 | findcard.6 | . . . . . . . . . 10 | |
40 | 24, 38, 39 | sylc 62 | . . . . . . . . 9 |
41 | 40 | 3exp 1180 | . . . . . . . 8 |
42 | 41 | alrimdv 1848 | . . . . . . 7 |
43 | breq1 3932 | . . . . . . . . 9 | |
44 | findcard.3 | . . . . . . . . 9 | |
45 | 43, 44 | imbi12d 233 | . . . . . . . 8 |
46 | 45 | cbvalv 1889 | . . . . . . 7 |
47 | 42, 46 | syl6ibr 161 | . . . . . 6 |
48 | 5, 8, 11, 17, 47 | finds1 4516 | . . . . 5 |
49 | 48 | 19.21bi 1537 | . . . 4 |
50 | 49 | rexlimiv 2543 | . . 3 |
51 | 2, 50 | sylbi 120 | . 2 |
52 | 1, 51 | vtoclga 2752 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wal 1329 wceq 1331 wcel 1480 wral 2416 wrex 2417 cvv 2686 cdif 3068 c0 3363 csn 3527 class class class wbr 3929 csuc 4287 com 4504 cen 6632 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-er 6429 df-en 6635 df-fin 6637 |
This theorem is referenced by: xpfi 6818 |
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