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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 6264 | . . 3 | |
3 | breq2 3789 | . . . . . . . 8 | |
4 | 3 | imbi1d 229 | . . . . . . 7 |
5 | 4 | albidv 1745 | . . . . . 6 |
6 | breq2 3789 | . . . . . . . 8 | |
7 | 6 | imbi1d 229 | . . . . . . 7 |
8 | 7 | albidv 1745 | . . . . . 6 |
9 | breq2 3789 | . . . . . . . 8 | |
10 | 9 | imbi1d 229 | . . . . . . 7 |
11 | 10 | albidv 1745 | . . . . . 6 |
12 | en0 6298 | . . . . . . . 8 | |
13 | findcard.5 | . . . . . . . . 9 | |
14 | findcard.1 | . . . . . . . . 9 | |
15 | 13, 14 | mpbiri 166 | . . . . . . . 8 |
16 | 12, 15 | sylbi 119 | . . . . . . 7 |
17 | 16 | ax-gen 1378 | . . . . . 6 |
18 | peano2 4336 | . . . . . . . . . . . . 13 | |
19 | breq2 3789 | . . . . . . . . . . . . . 14 | |
20 | 19 | rspcev 2701 | . . . . . . . . . . . . 13 |
21 | 18, 20 | sylan 277 | . . . . . . . . . . . 12 |
22 | isfi 6264 | . . . . . . . . . . . 12 | |
23 | 21, 22 | sylibr 132 | . . . . . . . . . . 11 |
24 | 23 | 3adant2 957 | . . . . . . . . . 10 |
25 | dif1en 6364 | . . . . . . . . . . . . . . . 16 | |
26 | 25 | 3expa 1138 | . . . . . . . . . . . . . . 15 |
27 | vex 2604 | . . . . . . . . . . . . . . . . 17 | |
28 | difexg 3919 | . . . . . . . . . . . . . . . . 17 | |
29 | 27, 28 | ax-mp 7 | . . . . . . . . . . . . . . . 16 |
30 | breq1 3788 | . . . . . . . . . . . . . . . . 17 | |
31 | findcard.2 | . . . . . . . . . . . . . . . . 17 | |
32 | 30, 31 | imbi12d 232 | . . . . . . . . . . . . . . . 16 |
33 | 29, 32 | spcv 2691 | . . . . . . . . . . . . . . 15 |
34 | 26, 33 | syl5com 29 | . . . . . . . . . . . . . 14 |
35 | 34 | ralrimdva 2441 | . . . . . . . . . . . . 13 |
36 | 35 | imp 122 | . . . . . . . . . . . 12 |
37 | 36 | an32s 532 | . . . . . . . . . . 11 |
38 | 37 | 3impa 1133 | . . . . . . . . . 10 |
39 | findcard.6 | . . . . . . . . . 10 | |
40 | 24, 38, 39 | sylc 61 | . . . . . . . . 9 |
41 | 40 | 3exp 1137 | . . . . . . . 8 |
42 | 41 | alrimdv 1797 | . . . . . . 7 |
43 | breq1 3788 | . . . . . . . . 9 | |
44 | findcard.3 | . . . . . . . . 9 | |
45 | 43, 44 | imbi12d 232 | . . . . . . . 8 |
46 | 45 | cbvalv 1835 | . . . . . . 7 |
47 | 42, 46 | syl6ibr 160 | . . . . . 6 |
48 | 5, 8, 11, 17, 47 | finds1 4343 | . . . . 5 |
49 | 48 | 19.21bi 1490 | . . . 4 |
50 | 49 | rexlimiv 2471 | . . 3 |
51 | 2, 50 | sylbi 119 | . 2 |
52 | 1, 51 | vtoclga 2664 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wal 1282 wceq 1284 wcel 1433 wral 2348 wrex 2349 cvv 2601 cdif 2970 c0 3251 csn 3398 class class class wbr 3785 csuc 4120 com 4331 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-coll 3893 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-fal 1290 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-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 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-res 4375 df-ima 4376 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-er 6129 df-en 6245 df-fin 6247 |
This theorem is referenced by: (None) |
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