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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 4145 |
. . . . . 6
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3 | finds.1 |
. . . . . 6
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4 | 2, 3 | elab 2896 |
. . . . 5
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5 | 1, 4 | mpbir 146 |
. . . 4
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6 | finds.6 |
. . . . . 6
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7 | vex 2755 |
. . . . . . 7
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8 | finds.2 |
. . . . . . 7
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9 | 7, 8 | elab 2896 |
. . . . . 6
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10 | 7 | sucex 4513 |
. . . . . . 7
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11 | finds.3 |
. . . . . . 7
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12 | 10, 11 | elab 2896 |
. . . . . 6
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13 | 6, 9, 12 | 3imtr4g 205 |
. . . . 5
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14 | 13 | rgen 2543 |
. . . 4
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15 | peano5 4612 |
. . . 4
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16 | 5, 14, 15 | mp2an 426 |
. . 3
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17 | 16 | sseli 3166 |
. 2
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18 | finds.4 |
. . 3
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19 | 18 | elabg 2898 |
. 2
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20 | 17, 19 | mpbid 147 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-iinf 4602 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-uni 3825 df-int 3860 df-suc 4386 df-iom 4605 |
This theorem is referenced by: findes 4617 nn0suc 4618 elomssom 4619 ordom 4621 nndceq0 4632 0elnn 4633 omsinds 4636 nna0r 6497 nnm0r 6498 nnsucelsuc 6510 nneneq 6875 php5 6876 php5dom 6881 fidcenumlemrk 6971 fidcenumlemr 6972 nnnninfeq 7144 nnnninfeq2 7145 frec2uzltd 10421 frecuzrdgg 10434 seq3val 10476 seqvalcd 10477 omgadd 10800 zfz1iso 10839 ennnfonelemhom 12434 nninfsellemdc 15157 |
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