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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn1m1nn 8857 | . . . 4 | |
2 | oveq1 5834 | . . . . . 6 | |
3 | 1m1e0 8908 | . . . . . 6 | |
4 | 2, 3 | eqtrdi 2206 | . . . . 5 |
5 | 4 | orim1i 750 | . . . 4 |
6 | 1, 5 | syl 14 | . . 3 |
7 | 6 | orcomd 719 | . 2 |
8 | elnn0 9098 | . 2 | |
9 | 7, 8 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wo 698 wceq 1335 wcel 2128 (class class class)co 5827 cc0 7735 c1 7736 cmin 8051 cn 8839 cn0 9096 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 ax-setind 4499 ax-cnex 7826 ax-resscn 7827 ax-1cn 7828 ax-1re 7829 ax-icn 7830 ax-addcl 7831 ax-addrcl 7832 ax-mulcl 7833 ax-addcom 7835 ax-addass 7837 ax-distr 7839 ax-i2m1 7840 ax-0id 7843 ax-rnegex 7844 ax-cnre 7846 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4029 df-id 4256 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-iota 5138 df-fun 5175 df-fv 5181 df-riota 5783 df-ov 5830 df-oprab 5831 df-mpo 5832 df-sub 8053 df-inn 8840 df-n0 9097 |
This theorem is referenced by: elnn0nn 9138 nnaddm1cl 9234 nn0n0n1ge2 9240 fseq1m1p1 10004 nn0ennn 10342 expm1t 10457 expgt1 10467 nn0ltexp2 10596 bcn1 10644 bcm1k 10646 bcn2m1 10655 resqrexlemnm 10930 resqrexlemcvg 10931 resqrexlemga 10935 binomlem 11392 arisum 11407 arisum2 11408 cvgratnnlemnexp 11433 cvgratnnlemfm 11438 mertenslem2 11445 iddvdsexp 11723 dvdsfac 11765 oexpneg 11781 phibnd 12108 phiprmpw 12113 prmdiv 12126 oddprm 12150 dvexp 13171 |
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