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Mirrors > Home > ILE Home > Th. List > nnap0 | GIF version |
Description: A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.) |
Ref | Expression |
---|---|
nnap0 | ⊢ (𝐴 ∈ ℕ → 𝐴 # 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnre 8943 | . 2 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
2 | nngt0 8961 | . 2 ⊢ (𝐴 ∈ ℕ → 0 < 𝐴) | |
3 | 1, 2 | gt0ap0d 8603 | 1 ⊢ (𝐴 ∈ ℕ → 𝐴 # 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2159 class class class wbr 4017 0cc0 7828 # cap 8555 ℕcn 8936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2161 ax-14 2162 ax-ext 2170 ax-sep 4135 ax-pow 4188 ax-pr 4223 ax-un 4447 ax-setind 4550 ax-cnex 7919 ax-resscn 7920 ax-1cn 7921 ax-1re 7922 ax-icn 7923 ax-addcl 7924 ax-addrcl 7925 ax-mulcl 7926 ax-mulrcl 7927 ax-addcom 7928 ax-mulcom 7929 ax-addass 7930 ax-mulass 7931 ax-distr 7932 ax-i2m1 7933 ax-0lt1 7934 ax-1rid 7935 ax-0id 7936 ax-rnegex 7937 ax-precex 7938 ax-cnre 7939 ax-pre-ltirr 7940 ax-pre-ltwlin 7941 ax-pre-lttrn 7942 ax-pre-apti 7943 ax-pre-ltadd 7944 ax-pre-mulgt0 7945 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-nel 2455 df-ral 2472 df-rex 2473 df-reu 2474 df-rab 2476 df-v 2753 df-sbc 2977 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-int 3859 df-br 4018 df-opab 4079 df-id 4307 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-iota 5192 df-fun 5232 df-fv 5238 df-riota 5846 df-ov 5893 df-oprab 5894 df-mpo 5895 df-pnf 8011 df-mnf 8012 df-xr 8013 df-ltxr 8014 df-le 8015 df-sub 8147 df-neg 8148 df-reap 8549 df-ap 8556 df-inn 8937 |
This theorem is referenced by: nndivre 8972 nndiv 8977 nndivtr 8978 nnap0d 8982 zdiv 9358 zdivadd 9359 zdivmul 9360 divfnzn 9638 qmulz 9640 qre 9642 qaddcl 9652 qnegcl 9653 qmulcl 9654 qapne 9656 nn0ledivnn 9784 flqdiv 10338 facdiv 10735 caucvgrelemcau 11006 expcnvap0 11527 ef0lem 11685 qredeq 12113 qredeu 12114 divgcdcoprm0 12118 isprm6 12164 sqrt2irr 12179 hashgcdlem 12255 pythagtriplem10 12286 pcqcl 12323 pcneg 12341 fldivp1 12363 infpnlem2 12375 rpcxproot 14717 |
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