Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > 3nn | GIF version |
Description: 3 is a positive integer. (Contributed by NM, 8-Jan-2006.) |
Ref | Expression |
---|---|
3nn | ⊢ 3 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3 8780 | . 2 ⊢ 3 = (2 + 1) | |
2 | 2nn 8881 | . . 3 ⊢ 2 ∈ ℕ | |
3 | peano2nn 8732 | . . 3 ⊢ (2 ∈ ℕ → (2 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (2 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2212 | 1 ⊢ 3 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 (class class class)co 5774 1c1 7621 + caddc 7623 ℕcn 8720 2c2 8771 3c3 8772 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-iota 5088 df-fv 5131 df-ov 5777 df-inn 8721 df-2 8779 df-3 8780 |
This theorem is referenced by: 4nn 8883 3nn0 8995 3z 9083 ige3m2fz 9829 sin01bnd 11464 3lcm2e6woprm 11767 3lcm2e6 11838 mulrndx 12069 mulrid 12070 mulrslid 12071 rngstrg 12074 tangtx 12919 |
Copyright terms: Public domain | W3C validator |