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Mirrors > Home > ILE Home > Th. List > nnzi | GIF version |
Description: A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
nnzi.1 | ⊢ 𝑁 ∈ ℕ |
Ref | Expression |
---|---|
nnzi | ⊢ 𝑁 ∈ ℤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnssz 9334 | . 2 ⊢ ℕ ⊆ ℤ | |
2 | nnzi.1 | . 2 ⊢ 𝑁 ∈ ℕ | |
3 | 1, 2 | sselii 3176 | 1 ⊢ 𝑁 ∈ ℤ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2164 ℕcn 8982 ℤcz 9317 |
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 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-addcom 7972 ax-addass 7974 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-0id 7980 ax-rnegex 7981 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-ltadd 7988 |
This theorem depends on definitions: df-bi 117 df-3or 981 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 2986 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-br 4030 df-opab 4091 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-iota 5215 df-fun 5256 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-inn 8983 df-z 9318 |
This theorem is referenced by: 1z 9343 2z 9345 3z 9346 4z 9347 3dvdsdec 12006 ndvdsi 12074 6gcd4e2 12132 3lcm2e6woprm 12224 6lcm4e12 12225 3lcm2e6 12298 prm23ge5 12402 pockthi 12496 strleun 12722 strle1g 12724 cnfldstr 14049 2logb9irr 15103 2logb9irrap 15109 lgsval 15120 lgsfvalg 15121 lgsfcl2 15122 lgsval2lem 15126 lgsdir2lem5 15148 lgsdir2 15149 lgsne0 15154 2lgsoddprmlem2 15194 ex-dvds 15222 ex-gcd 15223 |
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