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Mirrors > Home > ILE Home > Th. List > 9nn | GIF version |
Description: 9 is a positive integer. (Contributed by NM, 21-Oct-2012.) |
Ref | Expression |
---|---|
9nn | ⊢ 9 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-9 8983 | . 2 ⊢ 9 = (8 + 1) | |
2 | 8nn 9084 | . . 3 ⊢ 8 ∈ ℕ | |
3 | peano2nn 8929 | . . 3 ⊢ (8 ∈ ℕ → (8 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (8 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2250 | 1 ⊢ 9 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2148 (class class class)co 5874 1c1 7811 + caddc 7813 ℕcn 8917 8c8 8974 9c9 8975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-sep 4121 ax-cnex 7901 ax-resscn 7902 ax-1re 7904 ax-addrcl 7907 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-iota 5178 df-fv 5224 df-ov 5877 df-inn 8918 df-2 8976 df-3 8977 df-4 8978 df-5 8979 df-6 8980 df-7 8981 df-8 8982 df-9 8983 |
This theorem is referenced by: 9nn0 9198 9p1e10 9384 10nn 9397 3dvdsdec 11864 tsetndx 12635 tsetid 12636 tsetslid 12637 tsetndxnn 12638 topgrpstrd 12645 eltpsg 13431 setsmsbasg 13872 2logb9irr 14282 sqrt2cxp2logb9e3 14286 2logb9irrap 14288 ex-gcd 14365 |
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