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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 8101 | . 2 ⊢ 5 = (4 + 1) | |
2 | 4nn 8195 | . . 3 ⊢ 4 ∈ ℕ | |
3 | peano2nn 8051 | . . 3 ⊢ (4 ∈ ℕ → (4 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 7 | . 2 ⊢ (4 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2151 | 1 ⊢ 5 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1433 (class class class)co 5532 1c1 6982 + caddc 6984 ℕcn 8039 4c4 8091 5c5 8092 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-cnex 7067 ax-resscn 7068 ax-1re 7070 ax-addrcl 7073 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535 df-inn 8040 df-2 8098 df-3 8099 df-4 8100 df-5 8101 |
This theorem is referenced by: 6nn 8197 5nn0 8308 |
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