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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 12331 | . 2 ⊢ 4 = (3 + 1) | |
| 2 | 3nn 12345 | . . 3 ⊢ 3 ∈ ℕ | |
| 3 | peano2nn 12278 | . . 3 ⊢ (3 ∈ ℕ → (3 + 1) ∈ ℕ) | |
| 4 | 2, 3 | ax-mp 5 | . 2 ⊢ (3 + 1) ∈ ℕ |
| 5 | 1, 4 | eqeltri 2837 | 1 ⊢ 4 ∈ ℕ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 (class class class)co 7431 1c1 11156 + caddc 11158 ℕcn 12266 3c3 12322 4c4 12323 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-1cn 11213 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 |
| This theorem is referenced by: 5nn 12352 4nn0 12545 4z 12651 fldiv4p1lem1div2 13875 fldiv4lem1div2 13877 iexpcyc 14246 fsumcube 16096 ef01bndlem 16220 flodddiv4 16452 6lcm4e12 16653 2expltfac 17130 8nprm 17149 37prm 17158 43prm 17159 83prm 17160 139prm 17161 631prm 17164 prmo4 17165 1259prm 17173 2503lem2 17175 starvndx 17346 starvid 17347 srngstr 17353 homndx 17455 homid 17456 slotsdifplendx2 17461 slotsdifocndx 17462 prdsvalstr 17497 catstr 18005 lt6abl 19913 pcoass 25057 minveclem3 25463 iblitg 25803 dveflem 26017 tan4thpiOLD 26557 atan1 26971 log2tlbnd 26988 log2ub 26992 bclbnd 27324 bpos1 27327 bposlem6 27333 bposlem7 27334 bposlem8 27335 bposlem9 27336 gausslemma2dlem4 27413 m1lgs 27432 2lgslem1a 27435 2lgslem3a 27440 2lgslem3b 27441 2lgslem3c 27442 2lgslem3d 27443 2sqreultlem 27491 2sqreunnltlem 27494 chebbnd1lem1 27513 chebbnd1lem2 27514 chebbnd1lem3 27515 pntibndlem1 27633 pntibndlem2 27635 pntibndlem3 27636 pntlema 27640 pntlemb 27641 pntlemg 27642 pntlemf 27649 upgr4cycl4dv4e 30204 fib5 34407 hgt750lem2 34667 hgt750leme 34673 iccioo01 37328 420gcd8e4 42007 420lcm8e840 42012 lcm4un 42017 lcmineqlem23 42052 lcmineqlem 42053 3lexlogpow5ineq2 42056 aks4d1p1p5 42076 rmydioph 43026 rmxdioph 43028 expdiophlem2 43034 inductionexd 44168 amgm4d 44213 257prm 47548 fmtno4sqrt 47558 fmtno4prmfac 47559 fmtno4prmfac193 47560 fmtno5nprm 47570 139prmALT 47583 mod42tp1mod8 47589 2exp340mod341 47720 341fppr2 47721 wtgoldbnnsum4prm 47789 bgoldbachlt 47800 tgblthelfgott 47802 prstclevalOLD 49158 prstcocvalOLD 49161 |
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