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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 8750 | . 2 ⊢ 5 = (4 + 1) | |
2 | 4nn 8851 | . . 3 ⊢ 4 ∈ ℕ | |
3 | peano2nn 8700 | . . 3 ⊢ (4 ∈ ℕ → (4 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (4 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2190 | 1 ⊢ 5 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1465 (class class class)co 5742 1c1 7589 + caddc 7591 ℕcn 8688 4c4 8741 5c5 8742 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-cnex 7679 ax-resscn 7680 ax-1re 7682 ax-addrcl 7685 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-iota 5058 df-fv 5101 df-ov 5745 df-inn 8689 df-2 8747 df-3 8748 df-4 8749 df-5 8750 |
This theorem is referenced by: 6nn 8853 5nn0 8965 scandx 12013 scaid 12014 scaslid 12015 lmodstrd 12019 ipsstrd 12027 |
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