Theorem lbreu 12216

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 5152	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑤))
2	1	rspcv 3618	. . . . . . 7 ⊢ (𝑤 ∈ 𝑆 → (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝑤))
3		breq2 5152	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑤 ≤ 𝑦 ↔ 𝑤 ≤ 𝑥))
4	3	rspcv 3618	. . . . . . 7 ⊢ (𝑥 ∈ 𝑆 → (∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦 → 𝑤 ≤ 𝑥))
5	2, 4	im2anan9r 621	. . . . . 6 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦) → (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥)))
6		ssel 3989	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ ℝ → (𝑥 ∈ 𝑆 → 𝑥 ∈ ℝ))
7		ssel 3989	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ ℝ → (𝑤 ∈ 𝑆 → 𝑤 ∈ ℝ))
8	6, 7	anim12d 609	. . . . . . . . . 10 ⊢ (𝑆 ⊆ ℝ → ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → (𝑥 ∈ ℝ ∧ 𝑤 ∈ ℝ)))
9	8	impcom 407	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ 𝑆 ⊆ ℝ) → (𝑥 ∈ ℝ ∧ 𝑤 ∈ ℝ))
10		letri3 11344	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑤 ∈ ℝ) → (𝑥 = 𝑤 ↔ (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥)))
11	9, 10	syl 17	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ 𝑆 ⊆ ℝ) → (𝑥 = 𝑤 ↔ (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥)))
12	11	exbiri 811	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → (𝑆 ⊆ ℝ → ((𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥) → 𝑥 = 𝑤)))
13	12	com23 86	. . . . . 6 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → ((𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥) → (𝑆 ⊆ ℝ → 𝑥 = 𝑤)))
14	5, 13	syld 47	. . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦) → (𝑆 ⊆ ℝ → 𝑥 = 𝑤)))
15	14	com3r 87	. . . 4 ⊢ (𝑆 ⊆ ℝ → ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦) → 𝑥 = 𝑤)))
16	15	ralrimivv 3198	. . 3 ⊢ (𝑆 ⊆ ℝ → ∀𝑥 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦) → 𝑥 = 𝑤))
17	16	anim1ci 616	. 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)))
18		breq1 5151	. . . 4 ⊢ (𝑥 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑦))
19	18	ralbidv 3176	. . 3 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦))
20	19	reu4 3740	. 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)))
21	17, 20	sylibr 234	1 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → ∃!𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  ∃!wreu 3376   ⊆ wss 3963   class class class wbr 5148  ℝcr 11152   ≤ cle 11294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-resscn 11210  ax-pre-lttri 11227  ax-pre-lttrn 11228

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299

This theorem is referenced by:  lbcl  12217  lble  12218  uzwo2  12952