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Mirrors > Home > ILE Home > Th. List > 9nn | GIF version |
Description: 9 is a positive integer. (Contributed by NM, 21-Oct-2012.) |
Ref | Expression |
---|---|
9nn | ⊢ 9 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-9 8779 | . 2 ⊢ 9 = (8 + 1) | |
2 | 8nn 8880 | . . 3 ⊢ 8 ∈ ℕ | |
3 | peano2nn 8725 | . . 3 ⊢ (8 ∈ ℕ → (8 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (8 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2210 | 1 ⊢ 9 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 (class class class)co 5767 1c1 7614 + caddc 7616 ℕcn 8713 8c8 8770 9c9 8771 |
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 2119 ax-sep 4041 ax-cnex 7704 ax-resscn 7705 ax-1re 7707 ax-addrcl 7710 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-iota 5083 df-fv 5126 df-ov 5770 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-5 8775 df-6 8776 df-7 8777 df-8 8778 df-9 8779 |
This theorem is referenced by: 9nn0 8994 9p1e10 9177 10nn 9190 3dvdsdec 11551 tsetndx 12096 tsetid 12097 tsetslid 12098 topgrpstrd 12099 eltpsg 12196 setsmsbasg 12637 ex-gcd 12932 |
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