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Mirrors > Home > MPE Home > Th. List > nn1suc | Structured version Visualization version Unicode version |
Description: If a statement holds for 1 and also holds for a successor, it holds for all positive integers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
nn1suc.1 |
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nn1suc.3 |
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nn1suc.4 |
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nn1suc.5 |
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nn1suc.6 |
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Ref | Expression |
---|---|
nn1suc |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn1suc.5 |
. . . . 5
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2 | 1ex 9656 |
. . . . . 6
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3 | nn1suc.1 |
. . . . . 6
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4 | 2, 3 | sbcie 3290 |
. . . . 5
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5 | 1, 4 | mpbir 214 |
. . . 4
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6 | 1nn 10642 |
. . . . . . 7
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7 | eleq1 2537 |
. . . . . . 7
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8 | 6, 7 | mpbiri 241 |
. . . . . 6
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9 | nn1suc.4 |
. . . . . . 7
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10 | 9 | sbcieg 3288 |
. . . . . 6
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11 | 8, 10 | syl 17 |
. . . . 5
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12 | dfsbcq 3257 |
. . . . 5
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13 | 11, 12 | bitr3d 263 |
. . . 4
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14 | 5, 13 | mpbiri 241 |
. . 3
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15 | 14 | a1i 11 |
. 2
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16 | ovex 6336 |
. . . . . 6
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17 | nn1suc.3 |
. . . . . 6
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18 | 16, 17 | sbcie 3290 |
. . . . 5
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19 | oveq1 6315 |
. . . . . 6
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20 | 19 | sbceq1d 3260 |
. . . . 5
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21 | 18, 20 | syl5bbr 267 |
. . . 4
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22 | nn1suc.6 |
. . . 4
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23 | 21, 22 | vtoclga 3099 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | nncn 10639 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | ax-1cn 9615 |
. . . . . 6
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26 | npcan 9904 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 24, 25, 26 | sylancl 675 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 27 | sbceq1d 3260 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 28, 10 | bitrd 261 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 23, 29 | syl5ib 227 |
. 2
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31 | nn1m1nn 10651 |
. 2
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32 | 15, 30, 31 | mpjaod 388 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 ax-un 6602 ax-resscn 9614 ax-1cn 9615 ax-icn 9616 ax-addcl 9617 ax-addrcl 9618 ax-mulcl 9619 ax-mulrcl 9620 ax-mulcom 9621 ax-addass 9622 ax-mulass 9623 ax-distr 9624 ax-i2m1 9625 ax-1ne0 9626 ax-1rid 9627 ax-rnegex 9628 ax-rrecex 9629 ax-cnre 9630 ax-pre-lttri 9631 ax-pre-lttrn 9632 ax-pre-ltadd 9633 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3or 1008 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-nel 2644 df-ral 2761 df-rex 2762 df-reu 2763 df-rab 2765 df-v 3033 df-sbc 3256 df-csb 3350 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-pss 3406 df-nul 3723 df-if 3873 df-pw 3944 df-sn 3960 df-pr 3962 df-tp 3964 df-op 3966 df-uni 4191 df-iun 4271 df-br 4396 df-opab 4455 df-mpt 4456 df-tr 4491 df-eprel 4750 df-id 4754 df-po 4760 df-so 4761 df-fr 4798 df-we 4800 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 df-pred 5387 df-ord 5433 df-on 5434 df-lim 5435 df-suc 5436 df-iota 5553 df-fun 5591 df-fn 5592 df-f 5593 df-f1 5594 df-fo 5595 df-f1o 5596 df-fv 5597 df-riota 6270 df-ov 6311 df-oprab 6312 df-mpt2 6313 df-om 6712 df-wrecs 7046 df-recs 7108 df-rdg 7146 df-er 7381 df-en 7588 df-dom 7589 df-sdom 7590 df-pnf 9695 df-mnf 9696 df-ltxr 9698 df-sub 9882 df-nn 10632 |
This theorem is referenced by: opsqrlem6 27879 |
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