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Mirrors > Home > ILE Home > Th. List > 8nn | 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 8899 | . 2 ⊢ 8 = (7 + 1) | |
2 | 7nn 9000 | . . 3 ⊢ 7 ∈ ℕ | |
3 | peano2nn 8846 | . . 3 ⊢ (7 ∈ ℕ → (7 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (7 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2230 | 1 ⊢ 8 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2128 (class class class)co 5825 1c1 7734 + caddc 7736 ℕcn 8834 7c7 8890 8c8 8891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 ax-sep 4083 ax-cnex 7824 ax-resscn 7825 ax-1re 7827 ax-addrcl 7830 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-br 3967 df-iota 5136 df-fv 5179 df-ov 5828 df-inn 8835 df-2 8893 df-3 8894 df-4 8895 df-5 8896 df-6 8897 df-7 8898 df-8 8899 |
This theorem is referenced by: 9nn 9002 8nn0 9114 ipndx 12370 ipid 12371 ipslid 12372 ipsstrd 12373 |
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