Theorem lbreu 12188

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 5146	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑤))
2	1	rspcv 3604	. . . . . . 7 ⊢ (𝑤 ∈ 𝑆 → (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝑤))
3		breq2 5146	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑤 ≤ 𝑦 ↔ 𝑤 ≤ 𝑥))
4	3	rspcv 3604	. . . . . . 7 ⊢ (𝑥 ∈ 𝑆 → (∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦 → 𝑤 ≤ 𝑥))
5	2, 4	im2anan9r 620	. . . . . 6 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦) → (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥)))
6		ssel 3971	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ ℝ → (𝑥 ∈ 𝑆 → 𝑥 ∈ ℝ))
7		ssel 3971	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ ℝ → (𝑤 ∈ 𝑆 → 𝑤 ∈ ℝ))
8	6, 7	anim12d 608	. . . . . . . . . 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 11323	. . . . . . . . 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 810	. . . . . . 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 3194	. . 3 ⊢ (𝑆 ⊆ ℝ → ∀𝑥 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦) → 𝑥 = 𝑤))
17	16	anim1ci 615	. 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 5145	. . . 4 ⊢ (𝑥 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑦))
19	18	ralbidv 3173	. . 3 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝑤 ≤ 𝑦))
20	19	reu4 3725	. 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 233	1 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → ∃!𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2099  ∀wral 3057  ∃wrex 3066  ∃!wreu 3370   ⊆ wss 3945   class class class wbr 5142  ℝcr 11131   ≤ cle 11273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-resscn 11189  ax-pre-lttri 11206  ax-pre-lttrn 11207

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-po 5584  df-so 5585  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-er 8718  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278

This theorem is referenced by:  lbcl  12189  lble  12190  uzwo2  12920