taupilemrplb - Mathbox for Jim Kingdon

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Theorem taupilemrplb 37315

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 11183	. 2 ⊢ 0 ∈ ℝ
2		inss1 4203	. . . . 5 ⊢ (ℝ⁺ ∩ 𝐴) ⊆ ℝ⁺
3	2	sseli 3945	. . . 4 ⊢ (𝑦 ∈ (ℝ⁺ ∩ 𝐴) → 𝑦 ∈ ℝ⁺)
4	3	rpge0d 13006	. . 3 ⊢ (𝑦 ∈ (ℝ⁺ ∩ 𝐴) → 0 ≤ 𝑦)
5	4	rgen 3047	. 2 ⊢ ∀𝑦 ∈ (ℝ⁺ ∩ 𝐴)0 ≤ 𝑦
6		breq1 5113	. . . 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 3591	. 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 3045 ∃wrex 3054 ∩ cin 3916 class class class wbr 5110 ℝcr 11074 0cc0 11075 ≤ cle 11216 ℝ⁺crp 12958

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 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-resscn 11132 ax-1cn 11133 ax-addrcl 11136 ax-rnegex 11146 ax-cnre 11148 ax-pre-lttri 11149

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-rp 12959

This theorem is referenced by: taupilem2 37317 taupi 37318