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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 11691 | . 2 ⊢ 4 = (3 + 1) | |
2 | 3nn 11705 | . . 3 ⊢ 3 ∈ ℕ | |
3 | peano2nn 11639 | . . 3 ⊢ (3 ∈ ℕ → (3 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (3 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2909 | 1 ⊢ 4 ∈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 (class class class)co 7145 1c1 10527 + caddc 10529 ℕcn 11627 3c3 11682 4c4 11683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-1cn 10584 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7148 df-om 7569 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 |
This theorem is referenced by: 5nn 11712 4nn0 11905 4z 12005 fldiv4p1lem1div2 13195 fldiv4lem1div2 13197 iexpcyc 13559 fsumcube 15404 ef01bndlem 15527 flodddiv4 15754 6lcm4e12 15950 2expltfac 16416 8nprm 16435 37prm 16444 43prm 16445 83prm 16446 139prm 16447 631prm 16450 prmo4 16451 1259prm 16459 2503lem2 16461 starvndx 16613 starvid 16614 ressstarv 16616 srngstr 16617 homndx 16677 homid 16678 resshom 16681 prdsvalstr 16716 oppchomfval 16974 oppcbas 16978 rescco 17092 catstr 17217 lt6abl 18946 pcoass 23557 minveclem3 23961 iblitg 24298 dveflem 24505 tan4thpi 25029 atan1 25433 log2tlbnd 25451 log2ub 25455 bclbnd 25784 bpos1 25787 bposlem6 25793 bposlem7 25794 bposlem8 25795 bposlem9 25796 gausslemma2dlem4 25873 m1lgs 25892 2lgslem1a 25895 2lgslem3a 25900 2lgslem3b 25901 2lgslem3c 25902 2lgslem3d 25903 2sqreultlem 25951 2sqreunnltlem 25954 chebbnd1lem1 25973 chebbnd1lem2 25974 chebbnd1lem3 25975 pntibndlem1 26093 pntibndlem2 26095 pntibndlem3 26096 pntlema 26100 pntlemb 26101 pntlemg 26102 pntlemf 26109 upgr4cycl4dv4e 27892 fib5 31563 hgt750lem2 31823 hgt750leme 31829 rmydioph 39491 rmxdioph 39493 expdiophlem2 39499 inductionexd 40385 amgm4d 40434 257prm 43570 fmtno4sqrt 43580 fmtno4prmfac 43581 fmtno4prmfac193 43582 fmtno5nprm 43592 139prmALT 43606 mod42tp1mod8 43614 2exp340mod341 43745 341fppr2 43746 wtgoldbnnsum4prm 43814 bgoldbachlt 43825 tgblthelfgott 43827 |
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