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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 8991 | . 2 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
2 | nngt0 9009 | . 2 ⊢ (𝐴 ∈ ℕ → 0 < 𝐴) | |
3 | 1, 2 | gt0ap0d 8650 | 1 ⊢ (𝐴 ∈ ℕ → 𝐴 # 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2164 class class class wbr 4030 0cc0 7874 # cap 8602 ℕcn 8984 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2987 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-br 4031 df-opab 4092 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-iota 5216 df-fun 5257 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-inn 8985 |
This theorem is referenced by: nndivre 9020 nndiv 9025 nndivtr 9026 nnap0d 9030 zdiv 9408 zdivadd 9409 zdivmul 9410 divfnzn 9689 qmulz 9691 qre 9693 qaddcl 9703 qnegcl 9704 qmulcl 9705 qapne 9707 nn0ledivnn 9836 flqdiv 10395 facdiv 10812 caucvgrelemcau 11127 expcnvap0 11648 ef0lem 11806 qredeq 12237 qredeu 12238 divgcdcoprm0 12242 isprm6 12288 sqrt2irr 12303 hashgcdlem 12379 pythagtriplem10 12410 pcqcl 12447 pcneg 12466 fldivp1 12489 infpnlem2 12501 znidomb 14157 rpcxproot 15089 |
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