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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 8810 | . 2 ⊢ 9 = (8 + 1) | |
2 | 8nn 8911 | . . 3 ⊢ 8 ∈ ℕ | |
3 | peano2nn 8756 | . . 3 ⊢ (8 ∈ ℕ → (8 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (8 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2213 | 1 ⊢ 9 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1481 (class class class)co 5782 1c1 7645 + caddc 7647 ℕcn 8744 8c8 8801 9c9 8802 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-cnex 7735 ax-resscn 7736 ax-1re 7738 ax-addrcl 7741 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-iota 5096 df-fv 5139 df-ov 5785 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-5 8806 df-6 8807 df-7 8808 df-8 8809 df-9 8810 |
This theorem is referenced by: 9nn0 9025 9p1e10 9208 10nn 9221 3dvdsdec 11598 tsetndx 12146 tsetid 12147 tsetslid 12148 topgrpstrd 12149 eltpsg 12246 setsmsbasg 12687 2logb9irr 13096 sqrt2cxp2logb9e3 13100 2logb9irrap 13102 ex-gcd 13114 |
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