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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 8896 | . . . 4 | |
2 | oveq1 5860 | . . . . . 6 | |
3 | 1m1e0 8947 | . . . . . 6 | |
4 | 2, 3 | eqtrdi 2219 | . . . . 5 |
5 | 4 | orim1i 755 | . . . 4 |
6 | 1, 5 | syl 14 | . . 3 |
7 | 6 | orcomd 724 | . 2 |
8 | elnn0 9137 | . 2 | |
9 | 7, 8 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wo 703 wceq 1348 wcel 2141 (class class class)co 5853 cc0 7774 c1 7775 cmin 8090 cn 8878 cn0 9135 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-sub 8092 df-inn 8879 df-n0 9136 |
This theorem is referenced by: elnn0nn 9177 nnaddm1cl 9273 nn0n0n1ge2 9282 fseq1m1p1 10051 nn0ennn 10389 expm1t 10504 expgt1 10514 nn0ltexp2 10644 bcn1 10692 bcm1k 10694 bcn2m1 10703 resqrexlemnm 10982 resqrexlemcvg 10983 resqrexlemga 10987 binomlem 11446 arisum 11461 arisum2 11462 cvgratnnlemnexp 11487 cvgratnnlemfm 11492 mertenslem2 11499 iddvdsexp 11777 dvdsfac 11820 oexpneg 11836 phibnd 12171 phiprmpw 12176 prmdiv 12189 oddprm 12213 fldivp1 12300 prmpwdvds 12307 dvexp 13469 lgslem1 13695 |
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