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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 11694 | . 2 ⊢ 7 = (6 + 1) | |
2 | 6nn 11715 | . . 3 ⊢ 6 ∈ ℕ | |
3 | peano2nn 11639 | . . 3 ⊢ (6 ∈ ℕ → (6 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (6 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2909 | 1 ⊢ 7 ∈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 (class class class)co 7145 1c1 10527 + caddc 10529 ℕcn 11627 6c6 11685 7c7 11686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-1cn 10584 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7148 df-om 7569 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 |
This theorem is referenced by: 8nn 11721 7nn0 11908 7prm 16434 17prm 16440 prmlem2 16443 37prm 16444 43prm 16445 83prm 16446 139prm 16447 163prm 16448 317prm 16449 631prm 16450 1259prm 16459 mcubic 25352 cubic2 25353 cubic 25354 quartlem1 25362 quartlem2 25363 log2ublem1 25452 log2ublem2 25453 log2ub 25455 lgsdir2lem3 25831 lngndx 26155 lngid 26157 ttgval 26589 ttglem 26590 eengstr 26694 ex-xp 28143 ex-mod 28156 ex-prmo 28166 hgt750lem2 31823 rmydioph 39491 expdiophlem2 39499 257prm 43570 fmtno5nprm 43592 139prmALT 43606 127prm 43610 8exp8mod9 43748 nnsum3primesle9 43806 bgoldbtbndlem1 43817 tgoldbach 43829 |
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