![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > unitssre | Structured version Visualization version GIF version |
Description: (0[,]1) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
unitssre | ⊢ (0[,]1) ⊆ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10078 | . 2 ⊢ 0 ∈ ℝ | |
2 | 1re 10077 | . 2 ⊢ 1 ∈ ℝ | |
3 | iccssre 12293 | . 2 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → (0[,]1) ⊆ ℝ) | |
4 | 1, 2, 3 | mp2an 708 | 1 ⊢ (0[,]1) ⊆ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2030 ⊆ wss 3607 (class class class)co 6690 ℝcr 9973 0cc0 9974 1c1 9975 [,]cicc 12216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-i2m1 10042 ax-1ne0 10043 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-icc 12220 |
This theorem is referenced by: rpnnen 15000 iitopon 22729 dfii2 22732 dfii3 22733 dfii5 22735 iirevcn 22776 iihalf1cn 22778 iihalf2cn 22780 iimulcn 22784 icchmeo 22787 xrhmeo 22792 icccvx 22796 lebnumii 22812 reparphti 22843 pcoass 22870 pcorevlem 22872 pcorev2 22874 pi1xfrcnv 22903 vitalilem1 23422 vitalilem4 23425 vitalilem5 23426 vitali 23427 dvlipcn 23802 abelth2 24241 chordthmlem4 24607 chordthmlem5 24608 leibpi 24714 cvxcl 24756 scvxcvx 24757 lgamgulmlem2 24801 ttgcontlem1 25810 axeuclidlem 25887 stcl 29203 unitsscn 30070 probun 30609 probvalrnd 30614 cvxpconn 31350 cvxsconn 31351 resconn 31354 cvmliftlem8 31400 poimirlem29 33568 poimirlem30 33569 poimirlem31 33570 poimir 33572 broucube 33573 k0004ss1 38766 k0004val0 38769 sqrlearg 40098 salgencntex 40879 |
Copyright terms: Public domain | W3C validator |