Theorem lbreu 8023

Description: If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.)

Assertion

Ref	Expression
lbreu	⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → ∃!𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)

Distinct variable group:   𝑥,𝑦,𝑆

Proof of Theorem lbreu

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 3789	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑤))
2	1	rspcv 2697	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑆 → (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝑤))
3		breq2 3789	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑤 ≤ 𝑦 ↔ 𝑤 ≤ 𝑥))
4	3	rspcv 2697	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑆 → (∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦 → 𝑤 ≤ 𝑥))
5	2, 4	im2anan9r 563	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦) → (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥)))
6		ssel 2993	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ ℝ → (𝑥 ∈ 𝑆 → 𝑥 ∈ ℝ))
7		ssel 2993	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ ℝ → (𝑤 ∈ 𝑆 → 𝑤 ∈ ℝ))
8	6, 7	anim12d 328	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ ℝ → ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → (𝑥 ∈ ℝ ∧ 𝑤 ∈ ℝ)))
9	8	impcom 123	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ 𝑆 ⊆ ℝ) → (𝑥 ∈ ℝ ∧ 𝑤 ∈ ℝ))
10		letri3 7192	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑤 ∈ ℝ) → (𝑥 = 𝑤 ↔ (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥)))
11	9, 10	syl 14	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ 𝑆 ⊆ ℝ) → (𝑥 = 𝑤 ↔ (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥)))
12	11	exbiri 374	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → (𝑆 ⊆ ℝ → ((𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥) → 𝑥 = 𝑤)))
13	12	com23 77	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → ((𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥) → (𝑆 ⊆ ℝ → 𝑥 = 𝑤)))
14	5, 13	syld 44	. . . . . 6 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦) → (𝑆 ⊆ ℝ → 𝑥 = 𝑤)))
15	14	com3r 78	. . . . 5 ⊢ (𝑆 ⊆ ℝ → ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦) → 𝑥 = 𝑤)))
16	15	ralrimivv 2442	. . . 4 ⊢ (𝑆 ⊆ ℝ → ∀𝑥 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦) → 𝑥 = 𝑤))
17	16	anim2i 334	. . 3 ⊢ ((∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ 𝑆 ⊆ ℝ) → (∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑥 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦) → 𝑥 = 𝑤)))
18	17	ancoms 264	. 2 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → (∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑥 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦) → 𝑥 = 𝑤)))
19		breq1 3788	. . . 4 ⊢ (𝑥 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑦))
20	19	ralbidv 2368	. . 3 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦))
21	20	reu4 2786	. 2 ⊢ (∃!𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ↔ (∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑥 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦) → 𝑥 = 𝑤)))
22	18, 21	sylibr 132	1 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → ∃!𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∈ wcel 1433  ∀wral 2348  ∃wrex 2349  ∃!wreu 2350   ⊆ wss 2973   class class class wbr 3785  ℝcr 6980   ≤ cle 7154

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-pre-ltirr 7088  ax-pre-apti 7091

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-xp 4369  df-cnv 4371  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159

This theorem is referenced by:  lbcl  8024  lble  8025