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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 8792 | . 2 ⊢ 8 = (7 + 1) | |
2 | 7nn 8893 | . . 3 ⊢ 7 ∈ ℕ | |
3 | peano2nn 8739 | . . 3 ⊢ (7 ∈ ℕ → (7 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (7 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2212 | 1 ⊢ 8 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 (class class class)co 5774 1c1 7628 + caddc 7630 ℕcn 8727 7c7 8783 8c8 8784 |
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 7718 ax-resscn 7719 ax-1re 7721 ax-addrcl 7724 |
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 8728 df-2 8786 df-3 8787 df-4 8788 df-5 8789 df-6 8790 df-7 8791 df-8 8792 |
This theorem is referenced by: 9nn 8895 8nn0 9007 ipndx 12107 ipid 12108 ipslid 12109 ipsstrd 12110 |
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