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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 interval is a subset of the set of the extended nonnegative reals. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 12-Dec-2016.) |
Ref | Expression |
---|---|
unitssxrge0 | ⊢ (0[,]1) ⊆ (0[,]+∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0e0iccpnf 13423 | . 2 ⊢ 0 ∈ (0[,]+∞) | |
2 | 1xr 11260 | . . 3 ⊢ 1 ∈ ℝ* | |
3 | 0le1 11724 | . . 3 ⊢ 0 ≤ 1 | |
4 | pnfge 13097 | . . . 4 ⊢ (1 ∈ ℝ* → 1 ≤ +∞) | |
5 | 2, 4 | ax-mp 5 | . . 3 ⊢ 1 ≤ +∞ |
6 | 0xr 11248 | . . . 4 ⊢ 0 ∈ ℝ* | |
7 | pnfxr 11255 | . . . 4 ⊢ +∞ ∈ ℝ* | |
8 | elicc1 13355 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (1 ∈ (0[,]+∞) ↔ (1 ∈ ℝ* ∧ 0 ≤ 1 ∧ 1 ≤ +∞))) | |
9 | 6, 7, 8 | mp2an 691 | . . 3 ⊢ (1 ∈ (0[,]+∞) ↔ (1 ∈ ℝ* ∧ 0 ≤ 1 ∧ 1 ≤ +∞)) |
10 | 2, 3, 5, 9 | mpbir3an 1342 | . 2 ⊢ 1 ∈ (0[,]+∞) |
11 | iccss2 13382 | . 2 ⊢ ((0 ∈ (0[,]+∞) ∧ 1 ∈ (0[,]+∞)) → (0[,]1) ⊆ (0[,]+∞)) | |
12 | 1, 10, 11 | mp2an 691 | 1 ⊢ (0[,]1) ⊆ (0[,]+∞) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ w3a 1088 ∈ wcel 2107 ⊆ wss 3946 class class class wbr 5144 (class class class)co 7396 0cc0 11097 1c1 11098 +∞cpnf 11232 ℝ*cxr 11234 ≤ cle 11236 [,]cicc 13314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-1st 7962 df-2nd 7963 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-icc 13318 |
This theorem is referenced by: probun 33349 |
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