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Mirrors > Home > ILE Home > Th. List > 4nn | GIF version |
Description: 4 is a positive integer. (Contributed by NM, 8-Jan-2006.) |
Ref | Expression |
---|---|
4nn | ⊢ 4 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-4 8895 | . 2 ⊢ 4 = (3 + 1) | |
2 | 3nn 8996 | . . 3 ⊢ 3 ∈ ℕ | |
3 | peano2nn 8846 | . . 3 ⊢ (3 ∈ ℕ → (3 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (3 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2230 | 1 ⊢ 4 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2128 (class class class)co 5825 1c1 7734 + caddc 7736 ℕcn 8834 3c3 8886 4c4 8887 |
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 |
This theorem is referenced by: 5nn 8998 4nn0 9110 4z 9198 fldiv4p1lem1div2 10208 iexpcyc 10527 resqrexlemnmsq 10921 ef01bndlem 11657 flodddiv4 11829 flodddiv4t2lthalf 11832 6lcm4e12 11968 starvndx 12351 starvid 12352 starvslid 12353 srngstrd 12354 dveflem 13129 tan4thpi 13204 |
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