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Mirrors > Home > ILE Home > Th. List > 3z | GIF version |
Description: 3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Ref | Expression |
---|---|
3z | ⊢ 3 ∈ ℤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3nn 9144 | . 2 ⊢ 3 ∈ ℕ | |
2 | 1 | nnzi 9338 | 1 ⊢ 3 ∈ ℤ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2164 3c3 9034 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-2 9041 df-3 9042 df-z 9318 |
This theorem is referenced by: fz0to4untppr 10190 4fvwrd4 10206 fzo0to3tp 10286 expnass 10716 ef01bndlem 11899 sin01bnd 11900 sin01gt0 11905 egt2lt3 11923 3dvdsdec 12006 3dvds2dec 12007 n2dvds3 12056 flodddiv4 12075 3lcm2e6woprm 12224 3prm 12266 oddprmge3 12273 2logb9irr 15103 2irrexpq 15108 2logb9irrap 15109 2irrexpqap 15110 lgsdir2lem5 15148 2lgsoddprmlem3 15199 ex-fl 15217 ex-ceil 15218 ex-bc 15221 ex-dvds 15222 ex-gcd 15223 |
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