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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > unitssxrge0 | Structured version Visualization version GIF version |
Description: The closed unit is a subset of the set of the extended nonnegative reals. Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 12-Dec-2016.) |
Ref | Expression |
---|---|
unitssxrge0 | ⊢ (0[,]1) ⊆ (0[,]+∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0e0iccpnf 12154 | . 2 ⊢ 0 ∈ (0[,]+∞) | |
2 | 1re 9918 | . . . 4 ⊢ 1 ∈ ℝ | |
3 | 2 | rexri 9976 | . . 3 ⊢ 1 ∈ ℝ* |
4 | 0le1 10430 | . . 3 ⊢ 0 ≤ 1 | |
5 | pnfge 11840 | . . . 4 ⊢ (1 ∈ ℝ* → 1 ≤ +∞) | |
6 | 3, 5 | ax-mp 5 | . . 3 ⊢ 1 ≤ +∞ |
7 | 0xr 9965 | . . . 4 ⊢ 0 ∈ ℝ* | |
8 | pnfxr 9971 | . . . 4 ⊢ +∞ ∈ ℝ* | |
9 | elicc1 12090 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (1 ∈ (0[,]+∞) ↔ (1 ∈ ℝ* ∧ 0 ≤ 1 ∧ 1 ≤ +∞))) | |
10 | 7, 8, 9 | mp2an 704 | . . 3 ⊢ (1 ∈ (0[,]+∞) ↔ (1 ∈ ℝ* ∧ 0 ≤ 1 ∧ 1 ≤ +∞)) |
11 | 3, 4, 6, 10 | mpbir3an 1237 | . 2 ⊢ 1 ∈ (0[,]+∞) |
12 | iccss2 12115 | . 2 ⊢ ((0 ∈ (0[,]+∞) ∧ 1 ∈ (0[,]+∞)) → (0[,]1) ⊆ (0[,]+∞)) | |
13 | 1, 11, 12 | mp2an 704 | 1 ⊢ (0[,]1) ⊆ (0[,]+∞) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ w3a 1031 ∈ wcel 1977 ⊆ wss 3540 class class class wbr 4583 (class class class)co 6549 0cc0 9815 1c1 9816 +∞cpnf 9950 ℝ*cxr 9952 ≤ cle 9954 [,]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-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
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-reu 2903 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-iun 4457 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-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 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-sub 10147 df-neg 10148 df-icc 12053 |
This theorem is referenced by: probun 29808 |
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