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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 11081 | . 2 ⊢ 4 = (3 + 1) | |
2 | 3nn 11186 | . . 3 ⊢ 3 ∈ ℕ | |
3 | peano2nn 11032 | . . 3 ⊢ (3 ∈ ℕ → (3 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (3 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2697 | 1 ⊢ 4 ∈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 (class class class)co 6650 1c1 9937 + caddc 9939 ℕcn 11020 3c3 11071 4c4 11072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-1cn 9994 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 |
This theorem is referenced by: 5nn 11188 4nn0 11311 4z 11411 fldiv4p1lem1div2 12636 fldiv4lem1div2 12638 iexpcyc 12969 fsumcube 14791 ef01bndlem 14914 flodddiv4 15137 6lcm4e12 15329 2expltfac 15799 8nprm 15818 37prm 15828 43prm 15829 83prm 15830 139prm 15831 631prm 15834 prmo4 15835 1259prm 15843 2503lem2 15845 starvndx 16004 starvid 16005 ressstarv 16007 srngfn 16008 homndx 16074 homid 16075 resshom 16078 prdsvalstr 16113 oppchomfval 16374 oppcbas 16378 rescco 16492 catstr 16617 lt6abl 18296 pcoass 22824 minveclem3 23200 iblitg 23535 dveflem 23742 tan4thpi 24266 atan1 24655 log2tlbnd 24672 log2ub 24676 bclbnd 25005 bpos1 25008 bposlem6 25014 bposlem7 25015 bposlem8 25016 bposlem9 25017 gausslemma2dlem4 25094 m1lgs 25113 2lgslem1a 25116 2lgslem3a 25121 2lgslem3b 25122 2lgslem3c 25123 2lgslem3d 25124 chebbnd1lem1 25158 chebbnd1lem2 25159 chebbnd1lem3 25160 pntibndlem1 25278 pntibndlem2 25280 pntibndlem3 25281 pntlema 25285 pntlemb 25286 pntlemg 25287 pntlemf 25294 upgr4cycl4dv4e 27045 fib5 30467 hgt750lem2 30730 hgt750leme 30736 rmydioph 37581 rmxdioph 37583 expdiophlem2 37589 inductionexd 38453 amgm4d 38503 257prm 41473 fmtno4sqrt 41483 fmtno4prmfac 41484 fmtno4prmfac193 41485 fmtno5nprm 41495 139prmALT 41511 mod42tp1mod8 41519 wtgoldbnnsum4prm 41690 bgoldbachlt 41701 tgblthelfgott 41703 bgoldbachltOLD 41707 tgblthelfgottOLD 41709 |
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