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Mirrors > Home > ILE Home > Th. List > 6nn | GIF version |
Description: 6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
6nn | ⊢ 6 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-6 9000 | . 2 ⊢ 6 = (5 + 1) | |
2 | 5nn 9101 | . . 3 ⊢ 5 ∈ ℕ | |
3 | peano2nn 8949 | . . 3 ⊢ (5 ∈ ℕ → (5 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (5 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2262 | 1 ⊢ 6 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 (class class class)co 5891 1c1 7830 + caddc 7832 ℕcn 8937 5c5 8991 6c6 8992 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-sep 4136 ax-cnex 7920 ax-resscn 7921 ax-1re 7923 ax-addrcl 7926 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-iota 5193 df-fv 5239 df-ov 5894 df-inn 8938 df-2 8996 df-3 8997 df-4 8998 df-5 8999 df-6 9000 |
This theorem is referenced by: 7nn 9103 6nn0 9215 ef01bndlem 11782 sin01bnd 11783 cos01bnd 11784 6gcd4e2 12014 6lcm4e12 12105 vscandx 12634 vscaid 12635 vscaslid 12640 lmodstrd 12641 ipsstrd 12653 sincos3rdpi 14661 pigt3 14662 ex-dvds 14879 ex-gcd 14880 |
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