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Mirrors > Home > MPE Home > Th. List > 7nn | Structured version Visualization version GIF version |
Description: 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
7nn | ⊢ 7 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-7 11895 | . 2 ⊢ 7 = (6 + 1) | |
2 | 6nn 11916 | . . 3 ⊢ 6 ∈ ℕ | |
3 | peano2nn 11839 | . . 3 ⊢ (6 ∈ ℕ → (6 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (6 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2834 | 1 ⊢ 7 ∈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 (class class class)co 7210 1c1 10727 + caddc 10729 ℕcn 11827 6c6 11886 7c7 11887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pr 5319 ax-un 7520 ax-1cn 10784 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-ral 3063 df-rex 3064 df-reu 3065 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-ov 7213 df-om 7642 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-nn 11828 df-2 11890 df-3 11891 df-4 11892 df-5 11893 df-6 11894 df-7 11895 |
This theorem is referenced by: 8nn 11922 7nn0 12109 7prm 16661 17prm 16667 prmlem2 16670 37prm 16671 43prm 16672 83prm 16673 139prm 16674 163prm 16675 317prm 16676 631prm 16677 1259prm 16686 mcubic 25727 cubic2 25728 cubic 25729 quartlem1 25737 quartlem2 25738 log2ublem1 25826 log2ublem2 25827 log2ub 25829 lgsdir2lem3 26205 lngndx 26529 lngid 26531 ttgval 26963 ttglem 26964 eengstr 27068 ex-xp 28516 ex-mod 28529 ex-prmo 28539 hgt750lem2 32341 60gcd7e1 39745 60lcm7e420 39750 lcm7un 39759 lcmineqlem 39792 3lexlogpow5ineq2 39795 3lexlogpow2ineq1 39798 3lexlogpow2ineq2 39799 rmydioph 40537 expdiophlem2 40545 257prm 44684 fmtno5nprm 44706 139prmALT 44719 127prm 44722 8exp8mod9 44859 nnsum3primesle9 44917 bgoldbtbndlem1 44928 tgoldbach 44940 |
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