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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 11496 | . 2 ⊢ ℕ0 = (ℕ ∪ {0}) | |
2 | nnssre 11226 | . . 3 ⊢ ℕ ⊆ ℝ | |
3 | 0re 10242 | . . . 4 ⊢ 0 ∈ ℝ | |
4 | snssi 4474 | . . . 4 ⊢ (0 ∈ ℝ → {0} ⊆ ℝ) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ {0} ⊆ ℝ |
6 | 2, 5 | unssi 3939 | . 2 ⊢ (ℕ ∪ {0}) ⊆ ℝ |
7 | 1, 6 | eqsstri 3784 | 1 ⊢ ℕ0 ⊆ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2145 ∪ cun 3721 ⊆ wss 3723 {csn 4316 ℝcr 10137 0cc0 10138 ℕcn 11222 ℕ0cn0 11495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-i2m1 10206 ax-1ne0 10207 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-ov 6795 df-om 7213 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-nn 11223 df-n0 11496 |
This theorem is referenced by: nn0sscn 11500 nn0re 11504 nn0rei 11506 nn0red 11555 ssnn0fi 12988 fsuppmapnn0fiublem 12993 fsuppmapnn0fiub 12994 hashxrcl 13346 ramtlecl 15907 ramcl2lem 15916 ramxrcl 15924 0ram2 15928 0ramcl 15930 mdegleb 24040 mdeglt 24041 mdegldg 24042 mdegxrcl 24043 mdegcl 24045 mdegaddle 24050 mdegmullem 24054 deg1mul3le 24092 plyeq0lem 24182 dgrval 24200 dgrcl 24205 dgrub 24206 dgrlb 24208 aannenlem2 24300 taylfval 24329 tgcgr4 25643 motcgrg 25656 dplti 29949 xrsmulgzz 30014 nn0omnd 30177 nn0archi 30179 esumcst 30461 oddpwdc 30752 breprexp 31047 lermxnn0 38040 hbtlem2 38217 ssnn0ssfz 42652 |
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