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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 3187 |
. 2
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2 | findcard2sd.a |
. . . 4
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3 | 2 | adantr 276 |
. . 3
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4 | sseq1 3190 |
. . . . . 6
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5 | 4 | anbi2d 464 |
. . . . 5
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6 | findcard2sd.ch |
. . . . 5
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7 | 5, 6 | imbi12d 234 |
. . . 4
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8 | sseq1 3190 |
. . . . . 6
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9 | 8 | anbi2d 464 |
. . . . 5
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10 | findcard2sd.th |
. . . . 5
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11 | 9, 10 | imbi12d 234 |
. . . 4
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12 | sseq1 3190 |
. . . . . 6
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13 | 12 | anbi2d 464 |
. . . . 5
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14 | findcard2sd.ta |
. . . . 5
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15 | 13, 14 | imbi12d 234 |
. . . 4
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16 | sseq1 3190 |
. . . . . 6
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17 | 16 | anbi2d 464 |
. . . . 5
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18 | findcard2sd.et |
. . . . 5
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19 | 17, 18 | imbi12d 234 |
. . . 4
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20 | findcard2sd.z |
. . . . 5
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21 | 20 | adantr 276 |
. . . 4
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22 | simprl 529 |
. . . . . . . 8
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23 | simprr 531 |
. . . . . . . . 9
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24 | 23 | unssad 3324 |
. . . . . . . 8
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25 | 22, 24 | jca 306 |
. . . . . . 7
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26 | simpll 527 |
. . . . . . . 8
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27 | id 19 |
. . . . . . . . . . 11
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28 | vsnid 3636 |
. . . . . . . . . . . 12
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29 | elun2 3315 |
. . . . . . . . . . . 12
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30 | 28, 29 | mp1i 10 |
. . . . . . . . . . 11
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31 | 27, 30 | sseldd 3168 |
. . . . . . . . . 10
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32 | 31 | ad2antll 491 |
. . . . . . . . 9
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33 | simplr 528 |
. . . . . . . . 9
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34 | 32, 33 | eldifd 3151 |
. . . . . . . 8
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35 | findcard2sd.i |
. . . . . . . 8
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36 | 22, 26, 24, 34, 35 | syl22anc 1249 |
. . . . . . 7
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37 | 25, 36 | embantd 56 |
. . . . . 6
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38 | 37 | ex 115 |
. . . . 5
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39 | 38 | com23 78 |
. . . 4
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40 | 7, 11, 15, 19, 21, 39 | findcard2s 6903 |
. . 3
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41 | 3, 40 | mpcom 36 |
. 2
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42 | 1, 41 | mpan2 425 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-iord 4378 df-on 4380 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-er 6548 df-en 6754 df-fin 6756 |
This theorem is referenced by: fimax2gtri 6914 finexdc 6915 unfidisj 6934 undifdc 6936 ssfirab 6946 fnfi 6949 dcfi 6993 difinfinf 7113 hashunlem 10797 hashxp 10819 fsumconst 11475 fsumrelem 11492 fprodcl2lem 11626 fprodconst 11641 fprodap0 11642 fprodrec 11650 fprodap0f 11657 fprodle 11661 fprodmodd 11662 iuncld 13886 fsumcncntop 14327 |
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