Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nn0p1gt0 | Structured version Visualization version GIF version |
Description: A nonnegative integer increased by 1 is greater than 0. (Contributed by Alexander van der Vekens, 3-Oct-2018.) |
Ref | Expression |
---|---|
nn0p1gt0 | ⊢ (𝑁 ∈ ℕ0 → 0 < (𝑁 + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0re 12242 | . 2 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
2 | 1red 10977 | . 2 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℝ) | |
3 | nn0ge0 12258 | . 2 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) | |
4 | 0lt1 11497 | . . 3 ⊢ 0 < 1 | |
5 | 4 | a1i 11 | . 2 ⊢ (𝑁 ∈ ℕ0 → 0 < 1) |
6 | 1, 2, 3, 5 | addgegt0d 11548 | 1 ⊢ (𝑁 ∈ ℕ0 → 0 < (𝑁 + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 class class class wbr 5079 (class class class)co 7271 0cc0 10872 1c1 10873 + caddc 10875 < clt 11010 ℕ0cn0 12233 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 |
This theorem is referenced by: ubmelm1fzo 13481 fi1uzind 14209 brfi1indALT 14212 ccatws1n0 14340 ccats1pfxeq 14425 chfacffsupp 22003 chfacfscmul0 22005 chfacfscmulgsum 22007 chfacfpmmul0 22009 chfacfpmmulgsum 22011 iswwlksnx 28201 wspn0 28285 rusgrnumwwlks 28335 wwlksext2clwwlk 28417 numclwwlk2lem1lem 28702 cycpmco2lem4 31392 cycpmco2lem5 31393 lighneallem4a 45029 blennngt2o2 45907 itcovalsuc 45982 ackvalsuc1mpt 45993 |
Copyright terms: Public domain | W3C validator |