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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 11260 | . 2 ⊢ 0 ∈ ℝ | |
2 | inss1 4244 | . . . . 5 ⊢ (ℝ+ ∩ 𝐴) ⊆ ℝ+ | |
3 | 2 | sseli 3990 | . . . 4 ⊢ (𝑦 ∈ (ℝ+ ∩ 𝐴) → 𝑦 ∈ ℝ+) |
4 | 3 | rpge0d 13078 | . . 3 ⊢ (𝑦 ∈ (ℝ+ ∩ 𝐴) → 0 ≤ 𝑦) |
5 | 4 | rgen 3060 | . 2 ⊢ ∀𝑦 ∈ (ℝ+ ∩ 𝐴)0 ≤ 𝑦 |
6 | breq1 5150 | . . . 4 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦)) | |
7 | 6 | ralbidv 3175 | . . 3 ⊢ (𝑥 = 0 → (∀𝑦 ∈ (ℝ+ ∩ 𝐴)𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ (ℝ+ ∩ 𝐴)0 ≤ 𝑦)) |
8 | 7 | rspcev 3621 | . 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 1536 ∈ wcel 2105 ∀wral 3058 ∃wrex 3067 ∩ cin 3961 class class class wbr 5147 ℝcr 11151 0cc0 11152 ≤ cle 11293 ℝ+crp 13031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-addrcl 11213 ax-rnegex 11223 ax-cnre 11225 ax-pre-lttri 11226 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-rp 13032 |
This theorem is referenced by: taupilem2 37304 taupi 37305 |
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