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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 12284 | . 2 ⊢ 4 = (3 + 1) | |
2 | 3nn 12298 | . . 3 ⊢ 3 ∈ ℕ | |
3 | peano2nn 12231 | . . 3 ⊢ (3 ∈ ℕ → (3 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (3 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2828 | 1 ⊢ 4 ∈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 (class class class)co 7412 1c1 11117 + caddc 11119 ℕcn 12219 3c3 12275 4c4 12276 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 ax-1cn 11174 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 |
This theorem is referenced by: 5nn 12305 4nn0 12498 4z 12603 fldiv4p1lem1div2 13807 fldiv4lem1div2 13809 iexpcyc 14178 fsumcube 16011 ef01bndlem 16134 flodddiv4 16363 6lcm4e12 16560 2expltfac 17033 8nprm 17052 37prm 17061 43prm 17062 83prm 17063 139prm 17064 631prm 17067 prmo4 17068 1259prm 17076 2503lem2 17078 starvndx 17254 starvid 17255 srngstr 17261 homndx 17363 homid 17364 slotsdifplendx2 17369 slotsdifocndx 17370 prdsvalstr 17405 oppchomfvalOLD 17666 oppcbasOLD 17671 resccoOLD 17788 catstr 17919 lt6abl 19811 pcoass 24871 minveclem3 25277 iblitg 25618 dveflem 25831 tan4thpi 26364 atan1 26774 log2tlbnd 26791 log2ub 26795 bclbnd 27127 bpos1 27130 bposlem6 27136 bposlem7 27137 bposlem8 27138 bposlem9 27139 gausslemma2dlem4 27216 m1lgs 27235 2lgslem1a 27238 2lgslem3a 27243 2lgslem3b 27244 2lgslem3c 27245 2lgslem3d 27246 2sqreultlem 27294 2sqreunnltlem 27297 chebbnd1lem1 27316 chebbnd1lem2 27317 chebbnd1lem3 27318 pntibndlem1 27436 pntibndlem2 27438 pntibndlem3 27439 pntlema 27443 pntlemb 27444 pntlemg 27445 pntlemf 27452 upgr4cycl4dv4e 29872 fib5 33869 hgt750lem2 34129 hgt750leme 34135 iccioo01 36674 420gcd8e4 41340 420lcm8e840 41345 lcm4un 41350 lcmineqlem23 41385 lcmineqlem 41386 3lexlogpow5ineq2 41389 aks4d1p1p5 41409 rmydioph 42218 rmxdioph 42220 expdiophlem2 42226 inductionexd 43371 amgm4d 43417 257prm 46690 fmtno4sqrt 46700 fmtno4prmfac 46701 fmtno4prmfac193 46702 fmtno5nprm 46712 139prmALT 46725 mod42tp1mod8 46731 2exp340mod341 46862 341fppr2 46863 wtgoldbnnsum4prm 46931 bgoldbachlt 46942 tgblthelfgott 46944 prstclevalOLD 47853 prstcocvalOLD 47856 |
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