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| Mirrors > Home > MPE Home > Th. List > 8nn | Structured version Visualization version GIF version | ||
| Description: 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) |
| Ref | Expression |
|---|---|
| 8nn | ⊢ 8 ∈ ℕ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-8 12335 | . 2 ⊢ 8 = (7 + 1) | |
| 2 | 7nn 12358 | . . 3 ⊢ 7 ∈ ℕ | |
| 3 | peano2nn 12278 | . . 3 ⊢ (7 ∈ ℕ → (7 + 1) ∈ ℕ) | |
| 4 | 2, 3 | ax-mp 5 | . 2 ⊢ (7 + 1) ∈ ℕ |
| 5 | 1, 4 | eqeltri 2837 | 1 ⊢ 8 ∈ ℕ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 (class class class)co 7431 1c1 11156 + caddc 11158 ℕcn 12266 7c7 12326 8c8 12327 |
| 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 df-8 12335 |
| This theorem is referenced by: 9nn 12364 8nn0 12549 37prm 17158 43prm 17159 83prm 17160 317prm 17163 1259lem4 17171 1259lem5 17172 2503prm 17177 4001prm 17182 ipndx 17374 ipid 17375 ipsstr 17380 phlstr 17390 tngipOLD 24667 quart1cl 26897 quart1lem 26898 quart1 26899 log2tlbnd 26988 bposlem8 27335 lgsdir2lem2 27370 lgsdir2lem3 27371 2lgslem3a1 27444 2lgslem3b1 27445 2lgslem3c1 27446 2lgslem3d1 27447 2lgslem4 27450 2lgsoddprmlem2 27453 pntlemr 27646 pntlemj 27647 edgfid 29005 edgfndx 29006 edgfndxnn 29007 edgfndxidOLD 29009 baseltedgfOLD 29011 ex-prmo 30478 hgt750lem 34666 hgt750lem2 34667 420gcd8e4 42007 420lcm8e840 42012 lcm8un 42021 lcmineqlem23 42052 lcmineqlem 42053 3lexlogpow5ineq2 42056 3lexlogpow2ineq1 42059 rmydioph 43026 fmtnoprmfac2lem1 47553 127prm 47586 mod42tp1mod8 47589 8even 47700 8exp8mod9 47723 9fppr8 47724 nfermltl8rev 47729 nfermltlrev 47731 nnsum4primesevenALTV 47788 wtgoldbnnsum4prm 47789 bgoldbnnsum3prm 47791 bgoldbtbndlem1 47792 tgblthelfgott 47802 tgoldbachlt 47803 |
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