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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 10645 | . 2 ⊢ 0 ∈ ℝ | |
2 | inss1 4207 | . . . . 5 ⊢ (ℝ+ ∩ 𝐴) ⊆ ℝ+ | |
3 | 2 | sseli 3965 | . . . 4 ⊢ (𝑦 ∈ (ℝ+ ∩ 𝐴) → 𝑦 ∈ ℝ+) |
4 | 3 | rpge0d 12438 | . . 3 ⊢ (𝑦 ∈ (ℝ+ ∩ 𝐴) → 0 ≤ 𝑦) |
5 | 4 | rgen 3150 | . 2 ⊢ ∀𝑦 ∈ (ℝ+ ∩ 𝐴)0 ≤ 𝑦 |
6 | breq1 5071 | . . . 4 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦)) | |
7 | 6 | ralbidv 3199 | . . 3 ⊢ (𝑥 = 0 → (∀𝑦 ∈ (ℝ+ ∩ 𝐴)𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ (ℝ+ ∩ 𝐴)0 ≤ 𝑦)) |
8 | 7 | rspcev 3625 | . 2 ⊢ ((0 ∈ ℝ ∧ ∀𝑦 ∈ (ℝ+ ∩ 𝐴)0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ+ ∩ 𝐴)𝑥 ≤ 𝑦) |
9 | 1, 5, 8 | mp2an 690 | 1 ⊢ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ+ ∩ 𝐴)𝑥 ≤ 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 ∩ cin 3937 class class class wbr 5068 ℝcr 10538 0cc0 10539 ≤ cle 10678 ℝ+crp 12392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-addrcl 10600 ax-rnegex 10610 ax-cnre 10612 ax-pre-lttri 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-rp 12393 |
This theorem is referenced by: taupilem2 34605 taupi 34606 |
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