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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 3087 | . 2 | |
2 | findcard2sd.a | . . . 4 | |
3 | 2 | adantr 274 | . . 3 |
4 | sseq1 3090 | . . . . . 6 | |
5 | 4 | anbi2d 459 | . . . . 5 |
6 | findcard2sd.ch | . . . . 5 | |
7 | 5, 6 | imbi12d 233 | . . . 4 |
8 | sseq1 3090 | . . . . . 6 | |
9 | 8 | anbi2d 459 | . . . . 5 |
10 | findcard2sd.th | . . . . 5 | |
11 | 9, 10 | imbi12d 233 | . . . 4 |
12 | sseq1 3090 | . . . . . 6 | |
13 | 12 | anbi2d 459 | . . . . 5 |
14 | findcard2sd.ta | . . . . 5 | |
15 | 13, 14 | imbi12d 233 | . . . 4 |
16 | sseq1 3090 | . . . . . 6 | |
17 | 16 | anbi2d 459 | . . . . 5 |
18 | findcard2sd.et | . . . . 5 | |
19 | 17, 18 | imbi12d 233 | . . . 4 |
20 | findcard2sd.z | . . . . 5 | |
21 | 20 | adantr 274 | . . . 4 |
22 | simprl 505 | . . . . . . . 8 | |
23 | simprr 506 | . . . . . . . . 9 | |
24 | 23 | unssad 3223 | . . . . . . . 8 |
25 | 22, 24 | jca 304 | . . . . . . 7 |
26 | simpll 503 | . . . . . . . 8 | |
27 | id 19 | . . . . . . . . . . 11 | |
28 | vsnid 3527 | . . . . . . . . . . . 12 | |
29 | elun2 3214 | . . . . . . . . . . . 12 | |
30 | 28, 29 | mp1i 10 | . . . . . . . . . . 11 |
31 | 27, 30 | sseldd 3068 | . . . . . . . . . 10 |
32 | 31 | ad2antll 482 | . . . . . . . . 9 |
33 | simplr 504 | . . . . . . . . 9 | |
34 | 32, 33 | eldifd 3051 | . . . . . . . 8 |
35 | findcard2sd.i | . . . . . . . 8 | |
36 | 22, 26, 24, 34, 35 | syl22anc 1202 | . . . . . . 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 6752 | . . 3 |
41 | 3, 40 | mpcom 36 | . 2 |
42 | 1, 41 | mpan2 421 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1316 wcel 1465 cdif 3038 cun 3039 wss 3041 c0 3333 csn 3497 cfn 6602 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-er 6397 df-en 6603 df-fin 6605 |
This theorem is referenced by: fimax2gtri 6763 finexdc 6764 unfidisj 6778 undifdc 6780 ssfirab 6790 fnfi 6793 difinfinf 6954 hashunlem 10518 hashxp 10540 fsumconst 11191 fsumrelem 11208 iuncld 12211 fsumcncntop 12652 |
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