taupilemrplb - Mathbox for Jim Kingdon

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Theorem taupilemrplb 33762

Description: A set of positive reals has (in the reals) a lower bound. (Contributed by Jim Kingdon, 19-Feb-2019.)

**Assertion**
Ref	Expression
taupilemrplb	⊢ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ⁺ ∩ 𝐴)𝑥 ≤ 𝑦

Distinct variable groups: 𝑥,𝑦 𝑥,𝐴 Allowed substitution hint: 𝐴(𝑦)

**Proof of Theorem taupilemrplb**
Step	Hyp	Ref	Expression
1		0re 10378	. 2 ⊢ 0 ∈ ℝ
2		inss1 4052	. . . . 5 ⊢ (ℝ⁺ ∩ 𝐴) ⊆ ℝ⁺
3	2	sseli 3816	. . . 4 ⊢ (𝑦 ∈ (ℝ⁺ ∩ 𝐴) → 𝑦 ∈ ℝ⁺)
4	3	rpge0d 12185	. . 3 ⊢ (𝑦 ∈ (ℝ⁺ ∩ 𝐴) → 0 ≤ 𝑦)
5	4	rgen 3103	. 2 ⊢ ∀𝑦 ∈ (ℝ⁺ ∩ 𝐴)0 ≤ 𝑦
6		breq1 4889	. . . 4 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦))
7	6	ralbidv 3167	. . 3 ⊢ (𝑥 = 0 → (∀𝑦 ∈ (ℝ⁺ ∩ 𝐴)𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ (ℝ⁺ ∩ 𝐴)0 ≤ 𝑦))
8	7	rspcev 3510	. 2 ⊢ ((0 ∈ ℝ ∧ ∀𝑦 ∈ (ℝ⁺ ∩ 𝐴)0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ⁺ ∩ 𝐴)𝑥 ≤ 𝑦)
9	1, 5, 8	mp2an 682	1 ⊢ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ⁺ ∩ 𝐴)𝑥 ≤ 𝑦

Colors of variables: wff setvar class

Syntax hints: = wceq 1601 ∈ wcel 2106 ∀wral 3089 ∃wrex 3090 ∩ cin 3790 class class class wbr 4886 ℝcr 10271 0cc0 10272 ≤ cle 10412 ℝ⁺crp 12137

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-addrcl 10333 ax-rnegex 10343 ax-cnre 10345 ax-pre-lttri 10346

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-rp 12138

This theorem is referenced by: taupilem2 33764 taupi 33765