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Mirrors > Home > ILE Home > Th. List > 10nn | GIF version |
Description: 10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
10nn | ⊢ ;10 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9p1e10 8879 | . 2 ⊢ (9 + 1) = ;10 | |
2 | 9nn 8584 | . . 3 ⊢ 9 ∈ ℕ | |
3 | peano2nn 8434 | . . 3 ⊢ (9 ∈ ℕ → (9 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 7 | . 2 ⊢ (9 + 1) ∈ ℕ |
5 | 1, 4 | eqeltrri 2161 | 1 ⊢ ;10 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1438 (class class class)co 5652 0cc0 7350 1c1 7351 + caddc 7353 ℕcn 8422 9c9 8480 ;cdc 8877 |
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 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-cnex 7436 ax-resscn 7437 ax-1cn 7438 ax-1re 7439 ax-icn 7440 ax-addcl 7441 ax-addrcl 7442 ax-mulcl 7443 ax-mulcom 7446 ax-addass 7447 ax-mulass 7448 ax-distr 7449 ax-1rid 7452 ax-0id 7453 ax-cnre 7456 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-br 3846 df-iota 4980 df-fv 5023 df-ov 5655 df-inn 8423 df-2 8481 df-3 8482 df-4 8483 df-5 8484 df-6 8485 df-7 8486 df-8 8487 df-9 8488 df-dec 8878 |
This theorem is referenced by: 10pos 8893 10re 8895 decnncl2 8900 declt 8904 decltc 8905 declti 8914 dec10p 8919 |
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