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| Mirrors > Home > MPE Home > Th. List > 5nn | Structured version Visualization version GIF version | ||
| Description: 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) |
| Ref | Expression |
|---|---|
| 5nn | ⊢ 5 ∈ ℕ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-5 12332 | . 2 ⊢ 5 = (4 + 1) | |
| 2 | 4nn 12349 | . . 3 ⊢ 4 ∈ ℕ | |
| 3 | peano2nn 12278 | . . 3 ⊢ (4 ∈ ℕ → (4 + 1) ∈ ℕ) | |
| 4 | 2, 3 | ax-mp 5 | . 2 ⊢ (4 + 1) ∈ ℕ |
| 5 | 1, 4 | eqeltri 2837 | 1 ⊢ 5 ∈ ℕ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 (class class class)co 7431 1c1 11156 + caddc 11158 ℕcn 12266 4c4 12323 5c5 12324 |
| 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 df-5 12332 |
| This theorem is referenced by: 6nn 12355 5nn0 12546 5eluz3 12927 5ndvds3 16450 5ndvds6 16451 prm23ge5 16853 dec5dvds 17102 dec5nprm 17104 dec2nprm 17105 5prm 17146 10nprm 17151 23prm 17156 prmlem2 17157 43prm 17159 83prm 17160 317prm 17163 prmo5 17166 scandx 17358 scaid 17359 lmodstr 17369 ipsstr 17380 ccondx 17457 ccoid 17458 slotsbhcdif 17459 slotsbhcdifOLD 17460 slotsdifplendx2 17461 slotsdifocndx 17462 prdsvalstr 17497 catstr 18005 lt6abl 19913 psrvalstr 21936 opsrscaOLD 22078 tngscaOLD 24663 log2ublem1 26989 log2ublem2 26990 log2ub 26992 birthday 26997 ppiublem1 27246 ppiublem2 27247 ppiub 27248 bclbnd 27324 bposlem3 27330 bposlem4 27331 bposlem5 27332 bposlem6 27333 bposlem8 27335 bposlem9 27336 lgsdir2lem3 27371 ex-eprel 30452 ex-xp 30455 fib6 34408 hgt750lem2 34667 hgt750leme 34673 12gcd5e1 42004 12lcm5e60 42009 lcm5un 42018 lcmineqlem 42053 3lexlogpow5ineq1 42055 3lexlogpow2ineq1 42059 3lexlogpow2ineq2 42060 3lexlogpow5ineq5 42061 aks4d1p1p6 42074 aks4d1p1 42077 rmydioph 43026 expdiophlem2 43034 algstr 43185 inductionexd 44168 mnringscadOLD 44242 plusmod5ne 47347 minusmod5ne 47351 minusmodnep2tmod 47355 257prm 47548 fmtno4prmfac193 47560 31prm 47584 41prothprm 47606 gbowge7 47750 gbege6 47752 stgoldbwt 47763 sbgoldbwt 47764 sbgoldbm 47771 sbgoldbo 47774 nnsum3primesle9 47781 gpg5order 48014 gpg5nbgrvtx13starlem1 48027 gpg5nbgrvtx13starlem2 48028 gpg5nbgrvtx13starlem3 48029 gpg5nbgr3star 48037 prstclevalOLD 49158 prstcocvalOLD 49161 |
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