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Mirrors > Home > ILE Home > Th. List > finds | Unicode version |
Description: Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. This is Metamath 100 proof #74. (Contributed by NM, 14-Apr-1995.) |
Ref | Expression |
---|---|
finds.1 |
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finds.2 |
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finds.3 |
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finds.4 |
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finds.5 |
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finds.6 |
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Ref | Expression |
---|---|
finds |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | finds.5 |
. . . . 5
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2 | 0ex 4013 |
. . . . . 6
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3 | finds.1 |
. . . . . 6
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4 | 2, 3 | elab 2796 |
. . . . 5
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5 | 1, 4 | mpbir 145 |
. . . 4
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6 | finds.6 |
. . . . . 6
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7 | vex 2658 |
. . . . . . 7
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8 | finds.2 |
. . . . . . 7
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9 | 7, 8 | elab 2796 |
. . . . . 6
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10 | 7 | sucex 4373 |
. . . . . . 7
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11 | finds.3 |
. . . . . . 7
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12 | 10, 11 | elab 2796 |
. . . . . 6
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13 | 6, 9, 12 | 3imtr4g 204 |
. . . . 5
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14 | 13 | rgen 2457 |
. . . 4
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15 | peano5 4470 |
. . . 4
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16 | 5, 14, 15 | mp2an 420 |
. . 3
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17 | 16 | sseli 3057 |
. 2
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18 | finds.4 |
. . 3
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19 | 18 | elabg 2797 |
. 2
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20 | 17, 19 | mpbid 146 |
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 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-iinf 4460 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-v 2657 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-pw 3476 df-sn 3497 df-pr 3498 df-uni 3701 df-int 3736 df-suc 4251 df-iom 4463 |
This theorem is referenced by: findes 4475 nn0suc 4476 elnn 4477 ordom 4478 nndceq0 4489 0elnn 4490 omsinds 4493 nna0r 6326 nnm0r 6327 nnsucelsuc 6339 nneneq 6702 php5 6703 php5dom 6708 fidcenumlemrk 6792 fidcenumlemr 6793 frec2uzltd 10063 frecuzrdgg 10076 seq3val 10118 seqvalcd 10119 omgadd 10435 zfz1iso 10471 ennnfonelemhom 11767 nninfalllemn 12883 nninfsellemdc 12887 |
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