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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 12837 | . 2 ⊢ 0 ∈ (0[,]+∞) | |
2 | 1xr 10689 | . . 3 ⊢ 1 ∈ ℝ* | |
3 | 0le1 11152 | . . 3 ⊢ 0 ≤ 1 | |
4 | pnfge 12513 | . . . 4 ⊢ (1 ∈ ℝ* → 1 ≤ +∞) | |
5 | 2, 4 | ax-mp 5 | . . 3 ⊢ 1 ≤ +∞ |
6 | 0xr 10677 | . . . 4 ⊢ 0 ∈ ℝ* | |
7 | pnfxr 10684 | . . . 4 ⊢ +∞ ∈ ℝ* | |
8 | elicc1 12770 | . . . 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 1338 | . 2 ⊢ 1 ∈ (0[,]+∞) |
11 | iccss2 12796 | . 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 209 ∧ w3a 1084 ∈ wcel 2111 ⊆ wss 3881 class class class wbr 5030 (class class class)co 7135 0cc0 10526 1c1 10527 +∞cpnf 10661 ℝ*cxr 10663 ≤ cle 10665 [,]cicc 12729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-icc 12733 |
This theorem is referenced by: probun 31787 |
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