Theorem poslubmo 18465

Description: Least upper bounds in a poset are unique if they exist. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by NM, 16-Jun-2017.)

Hypotheses

Ref	Expression
poslubmo.l	⊢ ≤ = (le‘𝐾)
poslubmo.b	⊢ 𝐵 = (Base‘𝐾)

Assertion

Ref	Expression
poslubmo	⊢ ((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) → ∃*𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))

Distinct variable groups:   𝑥, ≤ ,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐾,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem poslubmo

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprrl 792	. . . . . 6 ⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤)
2		breq2 5115	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑦 ≤ 𝑧 ↔ 𝑦 ≤ 𝑤))
3	2	ralbidv 3194	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 ↔ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤))
4		breq2 5115	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (𝑥 ≤ 𝑧 ↔ 𝑥 ≤ 𝑤))
5	3, 4	imbi12d 347	. . . . . . 7 ⊢ (𝑧 = 𝑤 → ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 → 𝑥 ≤ 𝑤)))
6		simprlr 791	. . . . . . 7 ⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))
7		simplrr 789	. . . . . . 7 ⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → 𝑤 ∈ 𝐵)
8	5, 6, 7	rspcdva 3589	. . . . . 6 ⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 → 𝑥 ≤ 𝑤))
9	1, 8	mpd 16	. . . . 5 ⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → 𝑥 ≤ 𝑤)
10		simprll 790	. . . . . 6 ⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥)
11		breq2 5115	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑦 ≤ 𝑧 ↔ 𝑦 ≤ 𝑥))
12	11	ralbidv 3194	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 ↔ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥))
13		breq2 5115	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑤 ≤ 𝑧 ↔ 𝑤 ≤ 𝑥))
14	12, 13	imbi12d 347	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 → 𝑤 ≤ 𝑥)))
15		simprrr 793	. . . . . . 7 ⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧))
16		simplrl 788	. . . . . . 7 ⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → 𝑥 ∈ 𝐵)
17	14, 15, 16	rspcdva 3589	. . . . . 6 ⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 → 𝑤 ≤ 𝑥))
18	10, 17	mpd 16	. . . . 5 ⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → 𝑤 ≤ 𝑥)
19		poslubmo.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
20		poslubmo.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
21	19, 20	posasymb 18375	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥) ↔ 𝑥 = 𝑤))
22	21	3expb 1136	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥) ↔ 𝑥 = 𝑤))
23	22	ad4ant13 763	. . . . 5 ⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → ((𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥) ↔ 𝑥 = 𝑤))
24	9, 18, 23	mpbi2and 724	. . . 4 ⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → 𝑥 = 𝑤)
25	24	ex 417	. . 3 ⊢ (((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧))) → 𝑥 = 𝑤))
26	25	ralrimivva 3214	. 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) → ∀𝑥 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧))) → 𝑥 = 𝑤))
27		breq2 5115	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 𝑤))
28	27	ralbidv 3194	. . . 4 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ↔ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤))
29		breq1 5114	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑥 ≤ 𝑧 ↔ 𝑤 ≤ 𝑧))
30	29	imbi2d 343	. . . . 5 ⊢ (𝑥 = 𝑤 → ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))
31	30	ralbidv 3194	. . . 4 ⊢ (𝑥 = 𝑤 → (∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))
32	28, 31	anbi12d 643	. . 3 ⊢ (𝑥 = 𝑤 → ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧))))
33	32	rmo4 3700	. 2 ⊢ (∃*𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧))) → 𝑥 = 𝑤))
34	26, 33	sylibr 237	1 ⊢ ((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) → ∃*𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃*wrmo 3374   ⊆ wss 3911   class class class wbr 5111  ‘cfv 6537  Basecbs 17269  lecple 17317  Posetcpo 18363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5271

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-mo 2573  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3375  df-rab 3423  df-v 3463  df-sbc 3752  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5112  df-iota 6493  df-fv 6545  df-proset 18350  df-poset 18369

This theorem is referenced by:  poslubd  18467  lubeldm2  49654