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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 13466 | . 2 ⊢ 0 ∈ (0[,]+∞) | |
2 | 1xr 11301 | . . 3 ⊢ 1 ∈ ℝ* | |
3 | 0le1 11765 | . . 3 ⊢ 0 ≤ 1 | |
4 | pnfge 13140 | . . . 4 ⊢ (1 ∈ ℝ* → 1 ≤ +∞) | |
5 | 2, 4 | ax-mp 5 | . . 3 ⊢ 1 ≤ +∞ |
6 | 0xr 11289 | . . . 4 ⊢ 0 ∈ ℝ* | |
7 | pnfxr 11296 | . . . 4 ⊢ +∞ ∈ ℝ* | |
8 | elicc1 13398 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (1 ∈ (0[,]+∞) ↔ (1 ∈ ℝ* ∧ 0 ≤ 1 ∧ 1 ≤ +∞))) | |
9 | 6, 7, 8 | mp2an 690 | . . 3 ⊢ (1 ∈ (0[,]+∞) ↔ (1 ∈ ℝ* ∧ 0 ≤ 1 ∧ 1 ≤ +∞)) |
10 | 2, 3, 5, 9 | mpbir3an 1338 | . 2 ⊢ 1 ∈ (0[,]+∞) |
11 | iccss2 13425 | . 2 ⊢ ((0 ∈ (0[,]+∞) ∧ 1 ∈ (0[,]+∞)) → (0[,]1) ⊆ (0[,]+∞)) | |
12 | 1, 10, 11 | mp2an 690 | 1 ⊢ (0[,]1) ⊆ (0[,]+∞) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ w3a 1084 ∈ wcel 2098 ⊆ wss 3940 class class class wbr 5143 (class class class)co 7415 0cc0 11136 1c1 11137 +∞cpnf 11273 ℝ*cxr 11275 ≤ cle 11277 [,]cicc 13357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 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 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-1st 7989 df-2nd 7990 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-icc 13361 |
This theorem is referenced by: probun 34095 |
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