Theorem dirge 18649

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 18646	. . . . . . 7 ⊢ (𝑅 ∈ DirRel → dom 𝑅 = ∪ ∪ 𝑅)
3	1, 2	eqtrid 2812	. . . . . 6 ⊢ (𝑅 ∈ DirRel → 𝑋 = ∪ ∪ 𝑅)
4	3	eleq2d 2851	. . . . 5 ⊢ (𝑅 ∈ DirRel → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ ∪ ∪ 𝑅))
5	3	eleq2d 2851	. . . . 5 ⊢ (𝑅 ∈ DirRel → (𝐵 ∈ 𝑋 ↔ 𝐵 ∈ ∪ ∪ 𝑅))
6	4, 5	anbi12d 643	. . . 4 ⊢ (𝑅 ∈ DirRel → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ↔ (𝐴 ∈ ∪ ∪ 𝑅 ∧ 𝐵 ∈ ∪ ∪ 𝑅)))
7		eqid 2765	. . . . . . . . 9 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅
8	7	isdir 18644	. . . . . . . 8 ⊢ (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅)))))
9	8	ibi 270	. . . . . . 7 ⊢ (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅))))
10	9	simprrd 785	. . . . . 6 ⊢ (𝑅 ∈ DirRel → (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅))
11		codir 6111	. . . . . 6 ⊢ ((∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅) ↔ ∀𝑦 ∈ ∪ ∪ 𝑅∀𝑧 ∈ ∪ ∪ 𝑅∃𝑥(𝑦𝑅𝑥 ∧ 𝑧𝑅𝑥))
12	10, 11	sylib 221	. . . . 5 ⊢ (𝑅 ∈ DirRel → ∀𝑦 ∈ ∪ ∪ 𝑅∀𝑧 ∈ ∪ ∪ 𝑅∃𝑥(𝑦𝑅𝑥 ∧ 𝑧𝑅𝑥))
13		breq1 5108	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦𝑅𝑥 ↔ 𝐴𝑅𝑥))
14	13	anbi1d 642	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝑦𝑅𝑥 ∧ 𝑧𝑅𝑥) ↔ (𝐴𝑅𝑥 ∧ 𝑧𝑅𝑥)))
15	14	exbidv 1944	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑦𝑅𝑥 ∧ 𝑧𝑅𝑥) ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑧𝑅𝑥)))
16		breq1 5108	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝑧𝑅𝑥 ↔ 𝐵𝑅𝑥))
17	16	anbi2d 641	. . . . . . 7 ⊢ (𝑧 = 𝐵 → ((𝐴𝑅𝑥 ∧ 𝑧𝑅𝑥) ↔ (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
18	17	exbidv 1944	. . . . . 6 ⊢ (𝑧 = 𝐵 → (∃𝑥(𝐴𝑅𝑥 ∧ 𝑧𝑅𝑥) ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
19	15, 18	rspc2v 3595	. . . . 5 ⊢ ((𝐴 ∈ ∪ ∪ 𝑅 ∧ 𝐵 ∈ ∪ ∪ 𝑅) → (∀𝑦 ∈ ∪ ∪ 𝑅∀𝑧 ∈ ∪ ∪ 𝑅∃𝑥(𝑦𝑅𝑥 ∧ 𝑧𝑅𝑥) → ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
20	12, 19	syl5com 32	. . . 4 ⊢ (𝑅 ∈ DirRel → ((𝐴 ∈ ∪ ∪ 𝑅 ∧ 𝐵 ∈ ∪ ∪ 𝑅) → ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
21	6, 20	sylbid 243	. . 3 ⊢ (𝑅 ∈ DirRel → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
22		reldir 18645	. . . . . . . . . 10 ⊢ (𝑅 ∈ DirRel → Rel 𝑅)
23		relelrn 5926	. . . . . . . . . 10 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝑥) → 𝑥 ∈ ran 𝑅)
24	22, 23	sylan 591	. . . . . . . . 9 ⊢ ((𝑅 ∈ DirRel ∧ 𝐴𝑅𝑥) → 𝑥 ∈ ran 𝑅)
25	24	ex 417	. . . . . . . 8 ⊢ (𝑅 ∈ DirRel → (𝐴𝑅𝑥 → 𝑥 ∈ ran 𝑅))
26		ssun2 4134	. . . . . . . . . . 11 ⊢ ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
27		dmrnssfld 5955	. . . . . . . . . . 11 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅
28	26, 27	sstri 3948	. . . . . . . . . 10 ⊢ ran 𝑅 ⊆ ∪ ∪ 𝑅
29	28, 3	sseqtrrid 3982	. . . . . . . . 9 ⊢ (𝑅 ∈ DirRel → ran 𝑅 ⊆ 𝑋)
30	29	sseld 3938	. . . . . . . 8 ⊢ (𝑅 ∈ DirRel → (𝑥 ∈ ran 𝑅 → 𝑥 ∈ 𝑋))
31	25, 30	syld 48	. . . . . . 7 ⊢ (𝑅 ∈ DirRel → (𝐴𝑅𝑥 → 𝑥 ∈ 𝑋))
32	31	adantrd 496	. . . . . 6 ⊢ (𝑅 ∈ DirRel → ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) → 𝑥 ∈ 𝑋))
33	32	ancrd 560	. . . . 5 ⊢ (𝑅 ∈ DirRel → ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) → (𝑥 ∈ 𝑋 ∧ (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))))
34	33	eximdv 1940	. . . 4 ⊢ (𝑅 ∈ DirRel → (∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) → ∃𝑥(𝑥 ∈ 𝑋 ∧ (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))))
35		df-rex 3090	. . . 4 ⊢ (∃𝑥 ∈ 𝑋 (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ↔ ∃𝑥(𝑥 ∈ 𝑋 ∧ (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
36	34, 35	imbitrrdi 255	. . 3 ⊢ (𝑅 ∈ DirRel → (∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) → ∃𝑥 ∈ 𝑋 (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
37	21, 36	syld 48	. 2 ⊢ (𝑅 ∈ DirRel → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
38	37	3impib 1132	1 ⊢ ((𝑅 ∈ DirRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089   ∪ cun 3905   ⊆ wss 3907  ∪ cuni 4868   class class class wbr 5105   I cid 5546   × cxp 5650  ^◡ccnv 5651  dom cdm 5652  ran crn 5653   ↾ cres 5654   ∘ ccom 5656  Rel wrel 5657  DirRelcdir 18640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-dir 18642

This theorem is referenced by:  tailfb  36750