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Mirrors > Home > ILE Home > Th. List > 5nn | GIF version |
Description: 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
5nn | ⊢ 5 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-5 7976 | . 2 ⊢ 5 = (4 + 1) | |
2 | 4nn 8079 | . . 3 ⊢ 4 ∈ ℕ | |
3 | peano2nn 7926 | . . 3 ⊢ (4 ∈ ℕ → (4 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 7 | . 2 ⊢ (4 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2110 | 1 ⊢ 5 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1393 (class class class)co 5512 1c1 6890 + caddc 6892 ℕcn 7914 4c4 7966 5c5 7967 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-cnex 6975 ax-resscn 6976 ax-1re 6978 ax-addrcl 6981 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 df-inn 7915 df-2 7973 df-3 7974 df-4 7975 df-5 7976 |
This theorem is referenced by: 6nn 8081 5nn0 8201 |
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