| Mathbox for Jim Kingdon |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > taupilemrplb | Structured version Visualization version GIF version | ||
| Description: A set of positive reals has (in the reals) a lower bound. (Contributed by Jim Kingdon, 19-Feb-2019.) |
| Ref | Expression |
|---|---|
| taupilemrplb | ⊢ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ+ ∩ 𝐴)𝑥 ≤ 𝑦 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0re 11263 | . 2 ⊢ 0 ∈ ℝ | |
| 2 | inss1 4237 | . . . . 5 ⊢ (ℝ+ ∩ 𝐴) ⊆ ℝ+ | |
| 3 | 2 | sseli 3979 | . . . 4 ⊢ (𝑦 ∈ (ℝ+ ∩ 𝐴) → 𝑦 ∈ ℝ+) |
| 4 | 3 | rpge0d 13081 | . . 3 ⊢ (𝑦 ∈ (ℝ+ ∩ 𝐴) → 0 ≤ 𝑦) |
| 5 | 4 | rgen 3063 | . 2 ⊢ ∀𝑦 ∈ (ℝ+ ∩ 𝐴)0 ≤ 𝑦 |
| 6 | breq1 5146 | . . . 4 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦)) | |
| 7 | 6 | ralbidv 3178 | . . 3 ⊢ (𝑥 = 0 → (∀𝑦 ∈ (ℝ+ ∩ 𝐴)𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ (ℝ+ ∩ 𝐴)0 ≤ 𝑦)) |
| 8 | 7 | rspcev 3622 | . 2 ⊢ ((0 ∈ ℝ ∧ ∀𝑦 ∈ (ℝ+ ∩ 𝐴)0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ+ ∩ 𝐴)𝑥 ≤ 𝑦) |
| 9 | 1, 5, 8 | mp2an 692 | 1 ⊢ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ+ ∩ 𝐴)𝑥 ≤ 𝑦 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ∩ cin 3950 class class class wbr 5143 ℝcr 11154 0cc0 11155 ≤ cle 11296 ℝ+crp 13034 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-addrcl 11216 ax-rnegex 11226 ax-cnre 11228 ax-pre-lttri 11229 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-rp 13035 |
| This theorem is referenced by: taupilem2 37323 taupi 37324 |
| Copyright terms: Public domain | W3C validator |