Theorem dirge 18321

Description: For any two elements of a directed set, there exists a third element greater than or equal to both. Note that this does not say that the two elements have a least upper bound. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Hypothesis

Ref	Expression
dirge.1	⊢ 𝑋 = dom 𝑅

Assertion

Ref	Expression
dirge	⊢ ((𝑅 ∈ DirRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑋

Proof of Theorem dirge

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dirge.1	. . . . . . 7 ⊢ 𝑋 = dom 𝑅
2		dirdm 18318	. . . . . . 7 ⊢ (𝑅 ∈ DirRel → dom 𝑅 = ∪ ∪ 𝑅)
3	1, 2	eqtrid 2790	. . . . . 6 ⊢ (𝑅 ∈ DirRel → 𝑋 = ∪ ∪ 𝑅)
4	3	eleq2d 2824	. . . . 5 ⊢ (𝑅 ∈ DirRel → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ ∪ ∪ 𝑅))
5	3	eleq2d 2824	. . . . 5 ⊢ (𝑅 ∈ DirRel → (𝐵 ∈ 𝑋 ↔ 𝐵 ∈ ∪ ∪ 𝑅))
6	4, 5	anbi12d 631	. . . 4 ⊢ (𝑅 ∈ DirRel → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ↔ (𝐴 ∈ ∪ ∪ 𝑅 ∧ 𝐵 ∈ ∪ ∪ 𝑅)))
7		eqid 2738	. . . . . . . . 9 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅
8	7	isdir 18316	. . . . . . . 8 ⊢ (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅)))))
9	8	ibi 266	. . . . . . 7 ⊢ (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅))))
10	9	simprrd 771	. . . . . 6 ⊢ (𝑅 ∈ DirRel → (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅))
11		codir 6025	. . . . . 6 ⊢ ((∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅) ↔ ∀𝑦 ∈ ∪ ∪ 𝑅∀𝑧 ∈ ∪ ∪ 𝑅∃𝑥(𝑦𝑅𝑥 ∧ 𝑧𝑅𝑥))
12	10, 11	sylib 217	. . . . 5 ⊢ (𝑅 ∈ DirRel → ∀𝑦 ∈ ∪ ∪ 𝑅∀𝑧 ∈ ∪ ∪ 𝑅∃𝑥(𝑦𝑅𝑥 ∧ 𝑧𝑅𝑥))
13		breq1 5077	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦𝑅𝑥 ↔ 𝐴𝑅𝑥))
14	13	anbi1d 630	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝑦𝑅𝑥 ∧ 𝑧𝑅𝑥) ↔ (𝐴𝑅𝑥 ∧ 𝑧𝑅𝑥)))
15	14	exbidv 1924	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑦𝑅𝑥 ∧ 𝑧𝑅𝑥) ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑧𝑅𝑥)))
16		breq1 5077	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝑧𝑅𝑥 ↔ 𝐵𝑅𝑥))
17	16	anbi2d 629	. . . . . . 7 ⊢ (𝑧 = 𝐵 → ((𝐴𝑅𝑥 ∧ 𝑧𝑅𝑥) ↔ (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
18	17	exbidv 1924	. . . . . 6 ⊢ (𝑧 = 𝐵 → (∃𝑥(𝐴𝑅𝑥 ∧ 𝑧𝑅𝑥) ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
19	15, 18	rspc2v 3570	. . . . 5 ⊢ ((𝐴 ∈ ∪ ∪ 𝑅 ∧ 𝐵 ∈ ∪ ∪ 𝑅) → (∀𝑦 ∈ ∪ ∪ 𝑅∀𝑧 ∈ ∪ ∪ 𝑅∃𝑥(𝑦𝑅𝑥 ∧ 𝑧𝑅𝑥) → ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
20	12, 19	syl5com 31	. . . 4 ⊢ (𝑅 ∈ DirRel → ((𝐴 ∈ ∪ ∪ 𝑅 ∧ 𝐵 ∈ ∪ ∪ 𝑅) → ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
21	6, 20	sylbid 239	. . 3 ⊢ (𝑅 ∈ DirRel → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
22		reldir 18317	. . . . . . . . . 10 ⊢ (𝑅 ∈ DirRel → Rel 𝑅)
23		relelrn 5854	. . . . . . . . . 10 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝑥) → 𝑥 ∈ ran 𝑅)
24	22, 23	sylan 580	. . . . . . . . 9 ⊢ ((𝑅 ∈ DirRel ∧ 𝐴𝑅𝑥) → 𝑥 ∈ ran 𝑅)
25	24	ex 413	. . . . . . . 8 ⊢ (𝑅 ∈ DirRel → (𝐴𝑅𝑥 → 𝑥 ∈ ran 𝑅))
26		ssun2 4107	. . . . . . . . . . 11 ⊢ ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
27		dmrnssfld 5879	. . . . . . . . . . 11 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅
28	26, 27	sstri 3930	. . . . . . . . . 10 ⊢ ran 𝑅 ⊆ ∪ ∪ 𝑅
29	28, 3	sseqtrrid 3974	. . . . . . . . 9 ⊢ (𝑅 ∈ DirRel → ran 𝑅 ⊆ 𝑋)
30	29	sseld 3920	. . . . . . . 8 ⊢ (𝑅 ∈ DirRel → (𝑥 ∈ ran 𝑅 → 𝑥 ∈ 𝑋))
31	25, 30	syld 47	. . . . . . 7 ⊢ (𝑅 ∈ DirRel → (𝐴𝑅𝑥 → 𝑥 ∈ 𝑋))
32	31	adantrd 492	. . . . . 6 ⊢ (𝑅 ∈ DirRel → ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) → 𝑥 ∈ 𝑋))
33	32	ancrd 552	. . . . 5 ⊢ (𝑅 ∈ DirRel → ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) → (𝑥 ∈ 𝑋 ∧ (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))))
34	33	eximdv 1920	. . . 4 ⊢ (𝑅 ∈ DirRel → (∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) → ∃𝑥(𝑥 ∈ 𝑋 ∧ (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))))
35		df-rex 3070	. . . 4 ⊢ (∃𝑥 ∈ 𝑋 (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ↔ ∃𝑥(𝑥 ∈ 𝑋 ∧ (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
36	34, 35	syl6ibr 251	. . 3 ⊢ (𝑅 ∈ DirRel → (∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) → ∃𝑥 ∈ 𝑋 (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
37	21, 36	syld 47	. 2 ⊢ (𝑅 ∈ DirRel → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
38	37	3impib 1115	1 ⊢ ((𝑅 ∈ DirRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ∪ cun 3885   ⊆ wss 3887  ∪ cuni 4839   class class class wbr 5074   I cid 5488   × cxp 5587  ^◡ccnv 5588  dom cdm 5589  ran crn 5590   ↾ cres 5591   ∘ ccom 5593  Rel wrel 5594  DirRelcdir 18312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-dir 18314

This theorem is referenced by:  tailfb  34566