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Mirrors > Home > ILE Home > Th. List > nn0n0n1ge2 | Unicode version |
Description: A nonnegative integer which is neither 0 nor 1 is greater than or equal to 2. (Contributed by Alexander van der Vekens, 6-Dec-2017.) |
Ref | Expression |
---|---|
nn0n0n1ge2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0cn 9145 | . . . . . 6 | |
2 | 1cnd 7936 | . . . . . 6 | |
3 | 1, 2, 2 | subsub4d 8261 | . . . . 5 |
4 | 1p1e2 8995 | . . . . . 6 | |
5 | 4 | oveq2i 5864 | . . . . 5 |
6 | 3, 5 | eqtr2di 2220 | . . . 4 |
7 | 6 | 3ad2ant1 1013 | . . 3 |
8 | 3simpa 989 | . . . . . . 7 | |
9 | elnnne0 9149 | . . . . . . 7 | |
10 | 8, 9 | sylibr 133 | . . . . . 6 |
11 | nnm1nn0 9176 | . . . . . 6 | |
12 | 10, 11 | syl 14 | . . . . 5 |
13 | 1, 2 | subeq0ad 8240 | . . . . . . . . 9 |
14 | 13 | biimpd 143 | . . . . . . . 8 |
15 | 14 | necon3d 2384 | . . . . . . 7 |
16 | 15 | imp 123 | . . . . . 6 |
17 | 16 | 3adant2 1011 | . . . . 5 |
18 | elnnne0 9149 | . . . . 5 | |
19 | 12, 17, 18 | sylanbrc 415 | . . . 4 |
20 | nnm1nn0 9176 | . . . 4 | |
21 | 19, 20 | syl 14 | . . 3 |
22 | 7, 21 | eqeltrd 2247 | . 2 |
23 | 2nn0 9152 | . . . . 5 | |
24 | 23 | jctl 312 | . . . 4 |
25 | 24 | 3ad2ant1 1013 | . . 3 |
26 | nn0sub 9278 | . . 3 | |
27 | 25, 26 | syl 14 | . 2 |
28 | 22, 27 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 973 wceq 1348 wcel 2141 wne 2340 class class class wbr 3989 (class class class)co 5853 cc0 7774 c1 7775 caddc 7777 cle 7955 cmin 8090 cn 8878 c2 8929 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-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 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-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3or 974 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-nel 2436 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-nul 3415 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-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-2 8937 df-n0 9136 df-z 9213 |
This theorem is referenced by: nn0n0n1ge2b 9291 |
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