taupilemrplb - Mathbox for Jim Kingdon

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Theorem taupilemrplb 37464

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 11132	. 2 ⊢ 0 ∈ ℝ
2		inss1 4187	. . . . 5 ⊢ (ℝ⁺ ∩ 𝐴) ⊆ ℝ⁺
3	2	sseli 3927	. . . 4 ⊢ (𝑦 ∈ (ℝ⁺ ∩ 𝐴) → 𝑦 ∈ ℝ⁺)
4	3	rpge0d 12951	. . 3 ⊢ (𝑦 ∈ (ℝ⁺ ∩ 𝐴) → 0 ≤ 𝑦)
5	4	rgen 3051	. 2 ⊢ ∀𝑦 ∈ (ℝ⁺ ∩ 𝐴)0 ≤ 𝑦
6		breq1 5099	. . . 4 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦))
7	6	ralbidv 3157	. . 3 ⊢ (𝑥 = 0 → (∀𝑦 ∈ (ℝ⁺ ∩ 𝐴)𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ (ℝ⁺ ∩ 𝐴)0 ≤ 𝑦))
8	7	rspcev 3574	. 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 1541 ∈ wcel 2113 ∀wral 3049 ∃wrex 3058 ∩ cin 3898 class class class wbr 5096 ℝcr 11023 0cc0 11024 ≤ cle 11165 ℝ⁺crp 12903

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-resscn 11081 ax-1cn 11082 ax-addrcl 11085 ax-rnegex 11095 ax-cnre 11097 ax-pre-lttri 11098

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-rp 12904

This theorem is referenced by: taupilem2 37466 taupi 37467