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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 9919 | . 2 ⊢ 0 ∈ ℝ | |
2 | 1re 9918 | . 2 ⊢ 1 ∈ ℝ | |
3 | iccssre 12126 | . 2 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → (0[,]1) ⊆ ℝ) | |
4 | 1, 2, 3 | mp2an 704 | 1 ⊢ (0[,]1) ⊆ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 ⊆ wss 3540 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 [,]cicc 12049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-icc 12053 |
This theorem is referenced by: rpnnen 14795 iitopon 22490 dfii2 22493 dfii3 22494 dfii5 22496 iirevcn 22537 iihalf1cn 22539 iihalf2cn 22541 iimulcn 22545 icchmeo 22548 xrhmeo 22553 icccvx 22557 lebnumii 22573 reparphti 22605 pcoass 22632 pcorevlem 22634 pcorev2 22636 pi1xfrcnv 22665 vitalilem1 23182 vitalilem1OLD 23183 vitalilem4 23186 vitalilem5 23187 vitali 23188 dvlipcn 23561 abelth2 24000 chordthmlem4 24362 chordthmlem5 24363 leibpi 24469 cvxcl 24511 scvxcvx 24512 lgamgulmlem2 24556 ttgcontlem1 25565 axeuclidlem 25642 stcl 28459 unitsscn 29270 probun 29808 probvalrnd 29813 cvxpcon 30478 cvxscon 30479 rescon 30482 cvmliftlem8 30528 poimirlem29 32608 poimirlem30 32609 poimirlem31 32610 poimir 32612 broucube 32613 k0004ss1 37469 k0004val0 37472 sqrlearg 38627 salgencntex 39237 |
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