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Mirrors > Home > ILE Home > Th. List > 10nn0 | GIF version |
Description: 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
10nn0 | ⊢ ;10 ∈ ℕ0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn0 8996 | . 2 ⊢ 1 ∈ ℕ0 | |
2 | 0nn0 8995 | . 2 ⊢ 0 ∈ ℕ0 | |
3 | 1, 2 | deccl 9199 | 1 ⊢ ;10 ∈ ℕ0 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 0cc0 7623 1c1 7624 ℕ0cn0 8980 ;cdc 9185 |
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-in1 603 ax-in2 604 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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-addcom 7723 ax-mulcom 7724 ax-addass 7725 ax-mulass 7726 ax-distr 7727 ax-i2m1 7728 ax-1rid 7730 ax-0id 7731 ax-rnegex 7732 ax-cnre 7734 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sub 7938 df-inn 8724 df-2 8782 df-3 8783 df-4 8784 df-5 8785 df-6 8786 df-7 8787 df-8 8788 df-9 8789 df-n0 8981 df-dec 9186 |
This theorem is referenced by: decnncl 9204 dec0u 9205 dec0h 9206 decsuc 9215 decle 9218 decma 9235 decmac 9236 decma2c 9237 decadd 9238 decaddc 9239 decsubi 9247 decmul1 9248 decmul1c 9249 decmul2c 9250 decmul10add 9253 9t11e99 9314 |
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