Theorem lbreu 8795

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 3965	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑤))
2	1	rspcv 2809	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑆 → (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝑤))
3		breq2 3965	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑤 ≤ 𝑦 ↔ 𝑤 ≤ 𝑥))
4	3	rspcv 2809	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑆 → (∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦 → 𝑤 ≤ 𝑥))
5	2, 4	im2anan9r 589	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦) → (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥)))
6		ssel 3118	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ ℝ → (𝑥 ∈ 𝑆 → 𝑥 ∈ ℝ))
7		ssel 3118	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ ℝ → (𝑤 ∈ 𝑆 → 𝑤 ∈ ℝ))
8	6, 7	anim12d 333	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ ℝ → ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → (𝑥 ∈ ℝ ∧ 𝑤 ∈ ℝ)))
9	8	impcom 124	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ 𝑆 ⊆ ℝ) → (𝑥 ∈ ℝ ∧ 𝑤 ∈ ℝ))
10		letri3 7937	. . . . . . . . . 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 380	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → (𝑆 ⊆ ℝ → ((𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥) → 𝑥 = 𝑤)))
13	12	com23 78	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → ((𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥) → (𝑆 ⊆ ℝ → 𝑥 = 𝑤)))
14	5, 13	syld 45	. . . . . 6 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦) → (𝑆 ⊆ ℝ → 𝑥 = 𝑤)))
15	14	com3r 79	. . . . 5 ⊢ (𝑆 ⊆ ℝ → ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦) → 𝑥 = 𝑤)))
16	15	ralrimivv 2535	. . . 4 ⊢ (𝑆 ⊆ ℝ → ∀𝑥 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦) → 𝑥 = 𝑤))
17	16	anim2i 340	. . 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 266	. 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 3964	. . . 4 ⊢ (𝑥 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑦))
20	19	ralbidv 2454	. . 3 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦))
21	20	reu4 2902	. 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 133	1 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → ∃!𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∈ wcel 2125  ∀wral 2432  ∃wrex 2433  ∃!wreu 2434   ⊆ wss 3098   class class class wbr 3961  ℝcr 7710   ≤ cle 7892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-cnex 7802  ax-resscn 7803  ax-pre-ltirr 7823  ax-pre-apti 7826

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-nel 2420  df-ral 2437  df-rex 2438  df-reu 2439  df-rmo 2440  df-rab 2441  df-v 2711  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-opab 4022  df-xp 4585  df-cnv 4587  df-pnf 7893  df-mnf 7894  df-xr 7895  df-ltxr 7896  df-le 7897

This theorem is referenced by:  lbcl  8796  lble  8797