taupilemrplb - Mathbox for Jim Kingdon

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Theorem taupilemrplb 37304

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 11117	. 2 ⊢ 0 ∈ ℝ
2		inss1 4188	. . . . 5 ⊢ (ℝ⁺ ∩ 𝐴) ⊆ ℝ⁺
3	2	sseli 3931	. . . 4 ⊢ (𝑦 ∈ (ℝ⁺ ∩ 𝐴) → 𝑦 ∈ ℝ⁺)
4	3	rpge0d 12941	. . 3 ⊢ (𝑦 ∈ (ℝ⁺ ∩ 𝐴) → 0 ≤ 𝑦)
5	4	rgen 3046	. 2 ⊢ ∀𝑦 ∈ (ℝ⁺ ∩ 𝐴)0 ≤ 𝑦
6		breq1 5095	. . . 4 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦))
7	6	ralbidv 3152	. . 3 ⊢ (𝑥 = 0 → (∀𝑦 ∈ (ℝ⁺ ∩ 𝐴)𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ (ℝ⁺ ∩ 𝐴)0 ≤ 𝑦))
8	7	rspcev 3577	. 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 2109 ∀wral 3044 ∃wrex 3053 ∩ cin 3902 class class class wbr 5092 ℝcr 11008 0cc0 11009 ≤ cle 11150 ℝ⁺crp 12893

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-resscn 11066 ax-1cn 11067 ax-addrcl 11070 ax-rnegex 11080 ax-cnre 11082 ax-pre-lttri 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-rp 12894

This theorem is referenced by: taupilem2 37306 taupi 37307