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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 6241 | . . 3 | |
3 | breq2 3768 | . . . . . . . 8 | |
4 | 3 | imbi1d 220 | . . . . . . 7 |
5 | 4 | albidv 1705 | . . . . . 6 |
6 | breq2 3768 | . . . . . . . 8 | |
7 | 6 | imbi1d 220 | . . . . . . 7 |
8 | 7 | albidv 1705 | . . . . . 6 |
9 | breq2 3768 | . . . . . . . 8 | |
10 | 9 | imbi1d 220 | . . . . . . 7 |
11 | 10 | albidv 1705 | . . . . . 6 |
12 | en0 6275 | . . . . . . . 8 | |
13 | findcard.5 | . . . . . . . . 9 | |
14 | findcard.1 | . . . . . . . . 9 | |
15 | 13, 14 | mpbiri 157 | . . . . . . . 8 |
16 | 12, 15 | sylbi 114 | . . . . . . 7 |
17 | 16 | ax-gen 1338 | . . . . . 6 |
18 | peano2 4318 | . . . . . . . . . . . . 13 | |
19 | breq2 3768 | . . . . . . . . . . . . . 14 | |
20 | 19 | rspcev 2656 | . . . . . . . . . . . . 13 |
21 | 18, 20 | sylan 267 | . . . . . . . . . . . 12 |
22 | isfi 6241 | . . . . . . . . . . . 12 | |
23 | 21, 22 | sylibr 137 | . . . . . . . . . . 11 |
24 | 23 | 3adant2 923 | . . . . . . . . . 10 |
25 | dif1en 6337 | . . . . . . . . . . . . . . . 16 | |
26 | 25 | 3expa 1104 | . . . . . . . . . . . . . . 15 |
27 | vex 2560 | . . . . . . . . . . . . . . . . 17 | |
28 | difexg 3898 | . . . . . . . . . . . . . . . . 17 | |
29 | 27, 28 | ax-mp 7 | . . . . . . . . . . . . . . . 16 |
30 | breq1 3767 | . . . . . . . . . . . . . . . . 17 | |
31 | findcard.2 | . . . . . . . . . . . . . . . . 17 | |
32 | 30, 31 | imbi12d 223 | . . . . . . . . . . . . . . . 16 |
33 | 29, 32 | spcv 2646 | . . . . . . . . . . . . . . 15 |
34 | 26, 33 | syl5com 26 | . . . . . . . . . . . . . 14 |
35 | 34 | ralrimdva 2399 | . . . . . . . . . . . . 13 |
36 | 35 | imp 115 | . . . . . . . . . . . 12 |
37 | 36 | an32s 502 | . . . . . . . . . . 11 |
38 | 37 | 3impa 1099 | . . . . . . . . . 10 |
39 | findcard.6 | . . . . . . . . . 10 | |
40 | 24, 38, 39 | sylc 56 | . . . . . . . . 9 |
41 | 40 | 3exp 1103 | . . . . . . . 8 |
42 | 41 | alrimdv 1756 | . . . . . . 7 |
43 | breq1 3767 | . . . . . . . . 9 | |
44 | findcard.3 | . . . . . . . . 9 | |
45 | 43, 44 | imbi12d 223 | . . . . . . . 8 |
46 | 45 | cbvalv 1794 | . . . . . . 7 |
47 | 42, 46 | syl6ibr 151 | . . . . . 6 |
48 | 5, 8, 11, 17, 47 | finds1 4325 | . . . . 5 |
49 | 48 | 19.21bi 1450 | . . . 4 |
50 | 49 | rexlimiv 2427 | . . 3 |
51 | 2, 50 | sylbi 114 | . 2 |
52 | 1, 51 | vtoclga 2619 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wal 1241 wceq 1243 wcel 1393 wral 2306 wrex 2307 cvv 2557 cdif 2914 c0 3224 csn 3375 class class class wbr 3764 csuc 4102 com 4313 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-coll 3872 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-fal 1249 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-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-if 3332 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 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-res 4357 df-ima 4358 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-er 6106 df-en 6222 df-fin 6224 |
This theorem is referenced by: (None) |
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