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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 |
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findcard2sd.th |
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findcard2sd.ta |
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findcard2sd.et |
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findcard2sd.z |
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findcard2sd.i |
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findcard2sd.a |
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Ref | Expression |
---|---|
findcard2sd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3067 |
. 2
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2 | findcard2sd.a |
. . . 4
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3 | 2 | adantr 272 |
. . 3
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4 | sseq1 3070 |
. . . . . 6
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5 | 4 | anbi2d 455 |
. . . . 5
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6 | findcard2sd.ch |
. . . . 5
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7 | 5, 6 | imbi12d 233 |
. . . 4
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8 | sseq1 3070 |
. . . . . 6
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9 | 8 | anbi2d 455 |
. . . . 5
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10 | findcard2sd.th |
. . . . 5
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11 | 9, 10 | imbi12d 233 |
. . . 4
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12 | sseq1 3070 |
. . . . . 6
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13 | 12 | anbi2d 455 |
. . . . 5
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14 | findcard2sd.ta |
. . . . 5
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15 | 13, 14 | imbi12d 233 |
. . . 4
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16 | sseq1 3070 |
. . . . . 6
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17 | 16 | anbi2d 455 |
. . . . 5
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18 | findcard2sd.et |
. . . . 5
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19 | 17, 18 | imbi12d 233 |
. . . 4
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20 | findcard2sd.z |
. . . . 5
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21 | 20 | adantr 272 |
. . . 4
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22 | simprl 501 |
. . . . . . . 8
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23 | simprr 502 |
. . . . . . . . 9
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24 | 23 | unssad 3200 |
. . . . . . . 8
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25 | 22, 24 | jca 302 |
. . . . . . 7
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26 | simpll 499 |
. . . . . . . 8
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27 | id 19 |
. . . . . . . . . . 11
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28 | vsnid 3504 |
. . . . . . . . . . . 12
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29 | elun2 3191 |
. . . . . . . . . . . 12
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30 | 28, 29 | mp1i 10 |
. . . . . . . . . . 11
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31 | 27, 30 | sseldd 3048 |
. . . . . . . . . 10
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32 | 31 | ad2antll 478 |
. . . . . . . . 9
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33 | simplr 500 |
. . . . . . . . 9
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34 | 32, 33 | eldifd 3031 |
. . . . . . . 8
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35 | findcard2sd.i |
. . . . . . . 8
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36 | 22, 26, 24, 34, 35 | syl22anc 1185 |
. . . . . . 7
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37 | 25, 36 | embantd 56 |
. . . . . 6
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38 | 37 | ex 114 |
. . . . 5
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39 | 38 | com23 78 |
. . . 4
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40 | 7, 11, 15, 19, 21, 39 | findcard2s 6713 |
. . 3
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41 | 3, 40 | mpcom 36 |
. 2
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42 | 1, 41 | mpan2 419 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440 |
This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-if 3422 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-id 4153 df-iord 4226 df-on 4228 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-er 6359 df-en 6565 df-fin 6567 |
This theorem is referenced by: fimax2gtri 6724 finexdc 6725 unfidisj 6739 undifdc 6741 ssfirab 6750 fnfi 6753 difinfinf 6901 hashunlem 10391 hashxp 10413 fsumconst 11062 fsumrelem 11079 iuncld 12066 |
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