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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 8994 | . 2 ⊢ 4 = (3 + 1) | |
2 | 3nn 9095 | . . 3 ⊢ 3 ∈ ℕ | |
3 | peano2nn 8945 | . . 3 ⊢ (3 ∈ ℕ → (3 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (3 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2260 | 1 ⊢ 4 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2158 (class class class)co 5888 1c1 7826 + caddc 7828 ℕcn 8933 3c3 8985 4c4 8986 |
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 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 ax-sep 4133 ax-cnex 7916 ax-resscn 7917 ax-1re 7919 ax-addrcl 7922 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-iota 5190 df-fv 5236 df-ov 5891 df-inn 8934 df-2 8992 df-3 8993 df-4 8994 |
This theorem is referenced by: 5nn 9097 4nn0 9209 4z 9297 fldiv4p1lem1div2 10319 iexpcyc 10639 resqrexlemnmsq 11040 ef01bndlem 11778 flodddiv4 11953 flodddiv4t2lthalf 11956 6lcm4e12 12101 starvndx 12612 starvid 12613 starvslid 12614 srngstrd 12619 homid 12702 homslid 12703 dveflem 14483 tan4thpi 14558 m1lgs 14748 |
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