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Mirrors > Home > MPE Home > Th. List > dfrp2 | Structured version Visualization version GIF version |
Description: Alternate definition of the positive real numbers. (Contributed by Thierry Arnoux, 4-May-2020.) |
Ref | Expression |
---|---|
dfrp2 | ⊢ ℝ+ = (0(,)+∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltpnf 13051 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞) | |
2 | 1 | adantr 482 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) → 𝑥 < +∞) |
3 | 2 | pm4.71i 561 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) ↔ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ 𝑥 < +∞)) |
4 | df-3an 1090 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 0 < 𝑥 ∧ 𝑥 < +∞) ↔ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ 𝑥 < +∞)) | |
5 | 3, 4 | bitr4i 278 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥 ∧ 𝑥 < +∞)) |
6 | elrp 12927 | . . 3 ⊢ (𝑥 ∈ ℝ+ ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) | |
7 | 0xr 11212 | . . . 4 ⊢ 0 ∈ ℝ* | |
8 | pnfxr 11219 | . . . 4 ⊢ +∞ ∈ ℝ* | |
9 | elioo2 13316 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑥 ∈ (0(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥 ∧ 𝑥 < +∞))) | |
10 | 7, 8, 9 | mp2an 691 | . . 3 ⊢ (𝑥 ∈ (0(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥 ∧ 𝑥 < +∞)) |
11 | 5, 6, 10 | 3bitr4i 303 | . 2 ⊢ (𝑥 ∈ ℝ+ ↔ 𝑥 ∈ (0(,)+∞)) |
12 | 11 | eqriv 2729 | 1 ⊢ ℝ+ = (0(,)+∞) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 class class class wbr 5111 (class class class)co 7363 ℝcr 11060 0cc0 11061 +∞cpnf 11196 ℝ*cxr 11198 < clt 11199 ℝ+crp 12925 (,)cioo 13275 |
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 2703 ax-sep 5262 ax-nul 5269 ax-pow 5326 ax-pr 5390 ax-un 7678 ax-cnex 11117 ax-resscn 11118 ax-1cn 11119 ax-addrcl 11122 ax-rnegex 11132 ax-cnre 11134 ax-pre-lttri 11135 ax-pre-lttrn 11136 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4289 df-if 4493 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4872 df-iun 4962 df-br 5112 df-opab 5174 df-mpt 5195 df-id 5537 df-po 5551 df-so 5552 df-xp 5645 df-rel 5646 df-cnv 5647 df-co 5648 df-dm 5649 df-rn 5650 df-res 5651 df-ima 5652 df-iota 6454 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-ov 7366 df-oprab 7367 df-mpo 7368 df-1st 7927 df-2nd 7928 df-er 8656 df-en 8892 df-dom 8893 df-sdom 8894 df-pnf 11201 df-mnf 11202 df-xr 11203 df-ltxr 11204 df-le 11205 df-rp 12926 df-ioo 13279 |
This theorem is referenced by: omssubadd 32990 aks4d1p1p6 40604 |
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