Theorem lbreu 10852

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 4587	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑤))
2	1	rspcv 3278	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑆 → (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝑤))
3		breq2 4587	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑤 ≤ 𝑦 ↔ 𝑤 ≤ 𝑥))
4	3	rspcv 3278	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑆 → (∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦 → 𝑤 ≤ 𝑥))
5	2, 4	im2anan9r 877	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦) → (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥)))
6		ssel 3562	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ ℝ → (𝑥 ∈ 𝑆 → 𝑥 ∈ ℝ))
7		ssel 3562	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ ℝ → (𝑤 ∈ 𝑆 → 𝑤 ∈ ℝ))
8	6, 7	anim12d 584	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ ℝ → ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → (𝑥 ∈ ℝ ∧ 𝑤 ∈ ℝ)))
9	8	impcom 445	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ 𝑆 ⊆ ℝ) → (𝑥 ∈ ℝ ∧ 𝑤 ∈ ℝ))
10		letri3 10002	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑤 ∈ ℝ) → (𝑥 = 𝑤 ↔ (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥)))
11	9, 10	syl 17	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ 𝑆 ⊆ ℝ) → (𝑥 = 𝑤 ↔ (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥)))
12	11	exbiri 650	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → (𝑆 ⊆ ℝ → ((𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥) → 𝑥 = 𝑤)))
13	12	com23 84	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → ((𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥) → (𝑆 ⊆ ℝ → 𝑥 = 𝑤)))
14	5, 13	syld 46	. . . . . 6 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦) → (𝑆 ⊆ ℝ → 𝑥 = 𝑤)))
15	14	com3r 85	. . . . 5 ⊢ (𝑆 ⊆ ℝ → ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦) → 𝑥 = 𝑤)))
16	15	ralrimivv 2953	. . . 4 ⊢ (𝑆 ⊆ ℝ → ∀𝑥 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦) → 𝑥 = 𝑤))
17	16	anim2i 591	. . 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 468	. 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 4586	. . . 4 ⊢ (𝑥 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑦))
20	19	ralbidv 2969	. . 3 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦))
21	20	reu4 3367	. 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 223	1 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → ∃!𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898   ⊆ wss 3540   class class class wbr 4583  ℝcr 9814   ≤ cle 9954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-pre-lttri 9889  ax-pre-lttrn 9890

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959

This theorem is referenced by:  lbcl  10853  lble  10854  uzwo2  11628