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| Description: Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.) | 
| Ref | Expression | 
|---|---|
| nn0ssre | ⊢ ℕ0 ⊆ ℝ | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-n0 12529 | . 2 ⊢ ℕ0 = (ℕ ∪ {0}) | |
| 2 | nnssre 12271 | . . 3 ⊢ ℕ ⊆ ℝ | |
| 3 | 0re 11264 | . . . 4 ⊢ 0 ∈ ℝ | |
| 4 | snssi 4807 | . . . 4 ⊢ (0 ∈ ℝ → {0} ⊆ ℝ) | |
| 5 | 3, 4 | ax-mp 5 | . . 3 ⊢ {0} ⊆ ℝ | 
| 6 | 2, 5 | unssi 4190 | . 2 ⊢ (ℕ ∪ {0}) ⊆ ℝ | 
| 7 | 1, 6 | eqsstri 4029 | 1 ⊢ ℕ0 ⊆ ℝ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2107 ∪ cun 3948 ⊆ wss 3950 {csn 4625 ℝcr 11155 0cc0 11156 ℕcn 12267 ℕ0cn0 12528 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-i2m1 11224 ax-1ne0 11225 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-nn 12268 df-n0 12529 | 
| This theorem is referenced by: nn0re 12537 nn0rei 12539 nn0red 12590 ssnn0fi 14027 fsuppmapnn0fiublem 14032 fsuppmapnn0fiub 14033 hashxrcl 14397 ramtlecl 17039 ramcl2lem 17048 ramxrcl 17056 0ram2 17060 0ramcl 17062 mdegleb 26104 mdeglt 26105 mdegldg 26106 mdegxrcl 26107 mdegcl 26109 mdegaddle 26114 mdegmullem 26118 deg1mul3le 26157 plyeq0lem 26250 dgrval 26268 dgrcl 26273 dgrub 26274 dgrlb 26276 aannenlem2 26372 taylfval 26401 tgcgr4 28540 motcgrg 28553 hashxpe 32812 dplti 32888 xrsmulgzz 33012 nn0omnd 33374 nn0archi 33376 esumcst 34065 oddpwdc 34357 breprexp 34649 lermxnn0 42967 hbtlem2 43141 ssnn0ssfz 48270 | 
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