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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 10958 | . 2 ⊢ 4 = (3 + 1) | |
2 | 3nn 11063 | . . 3 ⊢ 3 ∈ ℕ | |
3 | peano2nn 10909 | . . 3 ⊢ (3 ∈ ℕ → (3 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (3 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2684 | 1 ⊢ 4 ∈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 (class class class)co 6549 1c1 9816 + caddc 9818 ℕcn 10897 3c3 10948 4c4 10949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-1cn 9873 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 |
This theorem is referenced by: 5nn 11065 4nn0 11188 4z 11288 fldiv4p1lem1div2 12498 fldiv4lem1div2 12500 iexpcyc 12831 fsumcube 14630 ef01bndlem 14753 flodddiv4 14975 6lcm4e12 15167 2expltfac 15637 8nprm 15656 37prm 15666 43prm 15667 83prm 15668 139prm 15669 631prm 15672 prmo4 15673 1259prm 15681 2503lem2 15683 starvndx 15827 starvid 15828 ressstarv 15830 srngfn 15831 homndx 15897 homid 15898 resshom 15901 prdsvalstr 15936 oppchomfval 16197 oppcbas 16201 rescco 16315 catstr 16440 lt6abl 18119 pcoass 22632 minveclem3 23008 iblitg 23341 dveflem 23546 tan4thpi 24070 atan1 24455 log2tlbnd 24472 log2ub 24476 bclbnd 24805 bpos1 24808 bposlem6 24814 bposlem7 24815 bposlem8 24816 bposlem9 24817 gausslemma2dlem4 24894 m1lgs 24913 2lgslem1a 24916 2lgslem3a 24921 2lgslem3b 24922 2lgslem3c 24923 2lgslem3d 24924 chebbnd1lem1 24958 chebbnd1lem2 24959 chebbnd1lem3 24960 pntibndlem1 25078 pntibndlem2 25080 pntibndlem3 25081 pntlema 25085 pntlemb 25086 pntlemg 25087 pntlemf 25094 4cycl4dv 26195 fib5 29794 rmydioph 36599 rmxdioph 36601 expdiophlem2 36607 inductionexd 37473 amgm4d 37525 257prm 40011 fmtno4sqrt 40021 fmtno4prmfac 40022 fmtno4prmfac193 40023 fmtno5nprm 40033 139prmALT 40049 mod42tp1mod8 40057 wtgoldbnnsum4prm 40218 bgoldbachlt 40227 tgblthelfgott 40229 bgoldbachltOLD 40234 tgblthelfgottOLD 40236 upgr4cycl4dv4e 41352 |
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