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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 8429 | . 2 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
2 | nngt0 8447 | . 2 ⊢ (𝐴 ∈ ℕ → 0 < 𝐴) | |
3 | 1, 2 | gt0ap0d 8105 | 1 ⊢ (𝐴 ∈ ℕ → 𝐴 # 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1438 class class class wbr 3845 0cc0 7350 # cap 8058 ℕcn 8422 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-cnex 7436 ax-resscn 7437 ax-1cn 7438 ax-1re 7439 ax-icn 7440 ax-addcl 7441 ax-addrcl 7442 ax-mulcl 7443 ax-mulrcl 7444 ax-addcom 7445 ax-mulcom 7446 ax-addass 7447 ax-mulass 7448 ax-distr 7449 ax-i2m1 7450 ax-0lt1 7451 ax-1rid 7452 ax-0id 7453 ax-rnegex 7454 ax-precex 7455 ax-cnre 7456 ax-pre-ltirr 7457 ax-pre-ltwlin 7458 ax-pre-lttrn 7459 ax-pre-apti 7460 ax-pre-ltadd 7461 ax-pre-mulgt0 7462 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-br 3846 df-opab 3900 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-iota 4980 df-fun 5017 df-fv 5023 df-riota 5608 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-pnf 7524 df-mnf 7525 df-xr 7526 df-ltxr 7527 df-le 7528 df-sub 7655 df-neg 7656 df-reap 8052 df-ap 8059 df-inn 8423 |
This theorem is referenced by: nndivre 8458 nndiv 8463 nndivtr 8464 nnap0d 8468 zdiv 8834 zdivadd 8835 zdivmul 8836 divfnzn 9106 qmulz 9108 qre 9110 qaddcl 9120 qnegcl 9121 qmulcl 9122 qapne 9124 nn0ledivnn 9238 flqdiv 9728 facdiv 10146 caucvgrelemcau 10413 expcnvap0 10896 ef0lem 10950 qredeq 11356 qredeu 11357 divgcdcoprm0 11361 isprm6 11404 sqrt2irr 11419 hashgcdlem 11481 |
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