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Mirrors > Home > MPE Home > Th. List > nn0ssre | Structured version Visualization version GIF version |
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 11886 | . 2 ⊢ ℕ0 = (ℕ ∪ {0}) | |
2 | nnssre 11630 | . . 3 ⊢ ℕ ⊆ ℝ | |
3 | 0re 10631 | . . . 4 ⊢ 0 ∈ ℝ | |
4 | snssi 4733 | . . . 4 ⊢ (0 ∈ ℝ → {0} ⊆ ℝ) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ {0} ⊆ ℝ |
6 | 2, 5 | unssi 4158 | . 2 ⊢ (ℕ ∪ {0}) ⊆ ℝ |
7 | 1, 6 | eqsstri 3998 | 1 ⊢ ℕ0 ⊆ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ∪ cun 3931 ⊆ wss 3933 {csn 4557 ℝcr 10524 0cc0 10525 ℕcn 11626 ℕ0cn0 11885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-i2m1 10593 ax-1ne0 10594 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-nn 11627 df-n0 11886 |
This theorem is referenced by: nn0re 11894 nn0rei 11896 nn0red 11944 ssnn0fi 13341 fsuppmapnn0fiublem 13346 fsuppmapnn0fiub 13347 hashxrcl 13706 ramtlecl 16324 ramcl2lem 16333 ramxrcl 16341 0ram2 16345 0ramcl 16347 mdegleb 24585 mdeglt 24586 mdegldg 24587 mdegxrcl 24588 mdegcl 24590 mdegaddle 24595 mdegmullem 24599 deg1mul3le 24637 plyeq0lem 24727 dgrval 24745 dgrcl 24750 dgrub 24751 dgrlb 24753 aannenlem2 24845 taylfval 24874 tgcgr4 26244 motcgrg 26257 hashxpe 30455 dplti 30508 xrsmulgzz 30592 nn0omnd 30841 nn0archi 30843 esumcst 31221 oddpwdc 31511 breprexp 31803 lermxnn0 39425 hbtlem2 39602 ssnn0ssfz 44325 |
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