taupilemrplb - Mathbox for Jim Kingdon

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Theorem taupilemrplb 33503

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 10246	. 2 ⊢ 0 ∈ ℝ
2		inss1 3981	. . . . 5 ⊢ (ℝ⁺ ∩ 𝐴) ⊆ ℝ⁺
3	2	sseli 3748	. . . 4 ⊢ (𝑦 ∈ (ℝ⁺ ∩ 𝐴) → 𝑦 ∈ ℝ⁺)
4	3	rpge0d 12079	. . 3 ⊢ (𝑦 ∈ (ℝ⁺ ∩ 𝐴) → 0 ≤ 𝑦)
5	4	rgen 3071	. 2 ⊢ ∀𝑦 ∈ (ℝ⁺ ∩ 𝐴)0 ≤ 𝑦
6		breq1 4790	. . . 4 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦))
7	6	ralbidv 3135	. . 3 ⊢ (𝑥 = 0 → (∀𝑦 ∈ (ℝ⁺ ∩ 𝐴)𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ (ℝ⁺ ∩ 𝐴)0 ≤ 𝑦))
8	7	rspcev 3460	. 2 ⊢ ((0 ∈ ℝ ∧ ∀𝑦 ∈ (ℝ⁺ ∩ 𝐴)0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ⁺ ∩ 𝐴)𝑥 ≤ 𝑦)
9	1, 5, 8	mp2an 672	1 ⊢ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ⁺ ∩ 𝐴)𝑥 ≤ 𝑦

Colors of variables: wff setvar class

Syntax hints: = wceq 1631 ∈ wcel 2145 ∀wral 3061 ∃wrex 3062 ∩ cin 3722 class class class wbr 4787 ℝcr 10141 0cc0 10142 ≤ cle 10281 ℝ⁺crp 12035

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-i2m1 10210 ax-1ne0 10211 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-ov 6799 df-er 7900 df-en 8114 df-dom 8115 df-sdom 8116 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-rp 12036

This theorem is referenced by: taupilem2 33505 taupi 33506