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Mirrors > Home > ILE Home > Th. List > findcard2sd | Unicode version |
Description: Deduction form of finite set induction . (Contributed by Jim Kingdon, 14-Sep-2021.) |
Ref | Expression |
---|---|
findcard2sd.ch | |
findcard2sd.th | |
findcard2sd.ta | |
findcard2sd.et | |
findcard2sd.z | |
findcard2sd.i | |
findcard2sd.a |
Ref | Expression |
---|---|
findcard2sd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3157 | . 2 | |
2 | findcard2sd.a | . . . 4 | |
3 | 2 | adantr 274 | . . 3 |
4 | sseq1 3160 | . . . . . 6 | |
5 | 4 | anbi2d 460 | . . . . 5 |
6 | findcard2sd.ch | . . . . 5 | |
7 | 5, 6 | imbi12d 233 | . . . 4 |
8 | sseq1 3160 | . . . . . 6 | |
9 | 8 | anbi2d 460 | . . . . 5 |
10 | findcard2sd.th | . . . . 5 | |
11 | 9, 10 | imbi12d 233 | . . . 4 |
12 | sseq1 3160 | . . . . . 6 | |
13 | 12 | anbi2d 460 | . . . . 5 |
14 | findcard2sd.ta | . . . . 5 | |
15 | 13, 14 | imbi12d 233 | . . . 4 |
16 | sseq1 3160 | . . . . . 6 | |
17 | 16 | anbi2d 460 | . . . . 5 |
18 | findcard2sd.et | . . . . 5 | |
19 | 17, 18 | imbi12d 233 | . . . 4 |
20 | findcard2sd.z | . . . . 5 | |
21 | 20 | adantr 274 | . . . 4 |
22 | simprl 521 | . . . . . . . 8 | |
23 | simprr 522 | . . . . . . . . 9 | |
24 | 23 | unssad 3294 | . . . . . . . 8 |
25 | 22, 24 | jca 304 | . . . . . . 7 |
26 | simpll 519 | . . . . . . . 8 | |
27 | id 19 | . . . . . . . . . . 11 | |
28 | vsnid 3602 | . . . . . . . . . . . 12 | |
29 | elun2 3285 | . . . . . . . . . . . 12 | |
30 | 28, 29 | mp1i 10 | . . . . . . . . . . 11 |
31 | 27, 30 | sseldd 3138 | . . . . . . . . . 10 |
32 | 31 | ad2antll 483 | . . . . . . . . 9 |
33 | simplr 520 | . . . . . . . . 9 | |
34 | 32, 33 | eldifd 3121 | . . . . . . . 8 |
35 | findcard2sd.i | . . . . . . . 8 | |
36 | 22, 26, 24, 34, 35 | syl22anc 1228 | . . . . . . 7 |
37 | 25, 36 | embantd 56 | . . . . . 6 |
38 | 37 | ex 114 | . . . . 5 |
39 | 38 | com23 78 | . . . 4 |
40 | 7, 11, 15, 19, 21, 39 | findcard2s 6847 | . . 3 |
41 | 3, 40 | mpcom 36 | . 2 |
42 | 1, 41 | mpan2 422 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1342 wcel 2135 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: fimax2gtri 6858 finexdc 6859 unfidisj 6878 undifdc 6880 ssfirab 6890 fnfi 6893 dcfi 6937 difinfinf 7057 hashunlem 10706 hashxp 10728 fsumconst 11381 fsumrelem 11398 fprodcl2lem 11532 fprodconst 11547 fprodap0 11548 fprodrec 11556 fprodap0f 11563 fprodle 11567 fprodmodd 11568 iuncld 12656 fsumcncntop 13097 |
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