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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 12334 | . 2 ⊢ 7 = (6 + 1) | |
| 2 | 6nn 12355 | . . 3 ⊢ 6 ∈ ℕ | |
| 3 | peano2nn 12278 | . . 3 ⊢ (6 ∈ ℕ → (6 + 1) ∈ ℕ) | |
| 4 | 2, 3 | ax-mp 5 | . 2 ⊢ (6 + 1) ∈ ℕ |
| 5 | 1, 4 | eqeltri 2837 | 1 ⊢ 7 ∈ ℕ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 (class class class)co 7431 1c1 11156 + caddc 11158 ℕcn 12266 6c6 12325 7c7 12326 |
| 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 df-6 12333 df-7 12334 |
| This theorem is referenced by: 8nn 12361 7nn0 12548 7prm 17148 17prm 17154 prmlem2 17157 37prm 17158 43prm 17159 83prm 17160 139prm 17161 163prm 17162 317prm 17163 631prm 17164 1259prm 17173 mcubic 26890 cubic2 26891 cubic 26892 quartlem1 26900 quartlem2 26901 log2ublem1 26989 log2ublem2 26990 log2ub 26992 lgsdir2lem3 27371 lngndx 28446 lngid 28448 slotslnbpsd 28450 lngndxnitvndx 28451 ttgvalOLD 28884 ttglemOLD 28886 eengstr 28995 ex-xp 30455 ex-mod 30468 ex-prmo 30478 hgt750lem2 34667 60gcd7e1 42006 60lcm7e420 42011 lcm7un 42020 lcmineqlem 42053 3lexlogpow5ineq2 42056 3lexlogpow2ineq1 42059 3lexlogpow2ineq2 42060 rmydioph 43026 expdiophlem2 43034 257prm 47548 fmtno5nprm 47570 139prmALT 47583 127prm 47586 8exp8mod9 47723 nnsum3primesle9 47781 bgoldbtbndlem1 47792 tgoldbach 47804 |
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