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Mirrors > Home > MPE Home > Th. List > 4nn | Structured version Visualization version GIF version |
Description: 4 is a positive integer. (Contributed by NM, 8-Jan-2006.) |
Ref | Expression |
---|---|
4nn | ⊢ 4 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-4 11119 | . 2 ⊢ 4 = (3 + 1) | |
2 | 3nn 11224 | . . 3 ⊢ 3 ∈ ℕ | |
3 | peano2nn 11070 | . . 3 ⊢ (3 ∈ ℕ → (3 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (3 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2726 | 1 ⊢ 4 ∈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2030 (class class class)co 6690 1c1 9975 + caddc 9977 ℕcn 11058 3c3 11109 4c4 11110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-1cn 10032 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 |
This theorem is referenced by: 5nn 11226 4nn0 11349 4z 11449 fldiv4p1lem1div2 12676 fldiv4lem1div2 12678 iexpcyc 13009 fsumcube 14835 ef01bndlem 14958 flodddiv4 15184 6lcm4e12 15376 2expltfac 15846 8nprm 15865 37prm 15875 43prm 15876 83prm 15877 139prm 15878 631prm 15881 prmo4 15882 1259prm 15890 2503lem2 15892 starvndx 16051 starvid 16052 ressstarv 16054 srngfn 16055 homndx 16121 homid 16122 resshom 16125 prdsvalstr 16160 oppchomfval 16421 oppcbas 16425 rescco 16539 catstr 16664 lt6abl 18342 pcoass 22870 minveclem3 23246 iblitg 23580 dveflem 23787 tan4thpi 24311 atan1 24700 log2tlbnd 24717 log2ub 24721 bclbnd 25050 bpos1 25053 bposlem6 25059 bposlem7 25060 bposlem8 25061 bposlem9 25062 gausslemma2dlem4 25139 m1lgs 25158 2lgslem1a 25161 2lgslem3a 25166 2lgslem3b 25167 2lgslem3c 25168 2lgslem3d 25169 chebbnd1lem1 25203 chebbnd1lem2 25204 chebbnd1lem3 25205 pntibndlem1 25323 pntibndlem2 25325 pntibndlem3 25326 pntlema 25330 pntlemb 25331 pntlemg 25332 pntlemf 25339 upgr4cycl4dv4e 27163 fib5 30595 hgt750lem2 30858 hgt750leme 30864 rmydioph 37898 rmxdioph 37900 expdiophlem2 37906 inductionexd 38770 amgm4d 38820 257prm 41798 fmtno4sqrt 41808 fmtno4prmfac 41809 fmtno4prmfac193 41810 fmtno5nprm 41820 139prmALT 41836 mod42tp1mod8 41844 wtgoldbnnsum4prm 42015 bgoldbachlt 42026 tgblthelfgott 42028 bgoldbachltOLD 42032 tgblthelfgottOLD 42034 |
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