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Mirrors > Home > ILE Home > Th. List > nnm1nn0 | Unicode version |
Description: A positive integer minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
nnm1nn0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn1m1nn 8440 |
. . . 4
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2 | oveq1 5659 |
. . . . . 6
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3 | 1m1e0 8491 |
. . . . . 6
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4 | 2, 3 | syl6eq 2136 |
. . . . 5
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5 | 4 | orim1i 712 |
. . . 4
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6 | 1, 5 | syl 14 |
. . 3
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7 | 6 | orcomd 683 |
. 2
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8 | elnn0 8675 |
. 2
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9 | 7, 8 | sylibr 132 |
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 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-setind 4353 ax-cnex 7436 ax-resscn 7437 ax-1cn 7438 ax-1re 7439 ax-icn 7440 ax-addcl 7441 ax-addrcl 7442 ax-mulcl 7443 ax-addcom 7445 ax-addass 7447 ax-distr 7449 ax-i2m1 7450 ax-0id 7453 ax-rnegex 7454 ax-cnre 7456 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-br 3846 df-opab 3900 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-iota 4980 df-fun 5017 df-fv 5023 df-riota 5608 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-sub 7655 df-inn 8423 df-n0 8674 |
This theorem is referenced by: elnn0nn 8715 nnaddm1cl 8811 nn0n0n1ge2 8817 fseq1m1p1 9509 nn0ennn 9840 expm1t 9983 expgt1 9993 bcn1 10166 bcm1k 10168 bcn2m1 10177 resqrexlemnm 10451 resqrexlemcvg 10452 resqrexlemga 10456 binomlem 10877 arisum 10892 arisum2 10893 cvgratnnlemnexp 10918 cvgratnnlemfm 10923 mertenslem2 10930 iddvdsexp 11098 dvdsfac 11139 oexpneg 11155 phibnd 11471 phiprmpw 11476 |
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