Theorem dirge 17590

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 17587	. . . . . . 7 ⊢ (𝑅 ∈ DirRel → dom 𝑅 = ∪ ∪ 𝑅)
3	1, 2	syl5eq 2873	. . . . . 6 ⊢ (𝑅 ∈ DirRel → 𝑋 = ∪ ∪ 𝑅)
4	3	eleq2d 2892	. . . . 5 ⊢ (𝑅 ∈ DirRel → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ ∪ ∪ 𝑅))
5	3	eleq2d 2892	. . . . 5 ⊢ (𝑅 ∈ DirRel → (𝐵 ∈ 𝑋 ↔ 𝐵 ∈ ∪ ∪ 𝑅))
6	4, 5	anbi12d 624	. . . 4 ⊢ (𝑅 ∈ DirRel → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ↔ (𝐴 ∈ ∪ ∪ 𝑅 ∧ 𝐵 ∈ ∪ ∪ 𝑅)))
7		eqid 2825	. . . . . . . . 9 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅
8	7	isdir 17585	. . . . . . . 8 ⊢ (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅)))))
9	8	ibi 259	. . . . . . 7 ⊢ (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅))))
10	9	simprrd 790	. . . . . 6 ⊢ (𝑅 ∈ DirRel → (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅))
11		codir 5758	. . . . . 6 ⊢ ((∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅) ↔ ∀𝑦 ∈ ∪ ∪ 𝑅∀𝑧 ∈ ∪ ∪ 𝑅∃𝑥(𝑦𝑅𝑥 ∧ 𝑧𝑅𝑥))
12	10, 11	sylib 210	. . . . 5 ⊢ (𝑅 ∈ DirRel → ∀𝑦 ∈ ∪ ∪ 𝑅∀𝑧 ∈ ∪ ∪ 𝑅∃𝑥(𝑦𝑅𝑥 ∧ 𝑧𝑅𝑥))
13		breq1 4876	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦𝑅𝑥 ↔ 𝐴𝑅𝑥))
14	13	anbi1d 623	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝑦𝑅𝑥 ∧ 𝑧𝑅𝑥) ↔ (𝐴𝑅𝑥 ∧ 𝑧𝑅𝑥)))
15	14	exbidv 2020	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑦𝑅𝑥 ∧ 𝑧𝑅𝑥) ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑧𝑅𝑥)))
16		breq1 4876	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝑧𝑅𝑥 ↔ 𝐵𝑅𝑥))
17	16	anbi2d 622	. . . . . . 7 ⊢ (𝑧 = 𝐵 → ((𝐴𝑅𝑥 ∧ 𝑧𝑅𝑥) ↔ (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
18	17	exbidv 2020	. . . . . 6 ⊢ (𝑧 = 𝐵 → (∃𝑥(𝐴𝑅𝑥 ∧ 𝑧𝑅𝑥) ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
19	15, 18	rspc2v 3539	. . . . 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 232	. . 3 ⊢ (𝑅 ∈ DirRel → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
22		reldir 17586	. . . . . . . . . 10 ⊢ (𝑅 ∈ DirRel → Rel 𝑅)
23		relelrn 5592	. . . . . . . . . 10 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝑥) → 𝑥 ∈ ran 𝑅)
24	22, 23	sylan 575	. . . . . . . . 9 ⊢ ((𝑅 ∈ DirRel ∧ 𝐴𝑅𝑥) → 𝑥 ∈ ran 𝑅)
25	24	ex 403	. . . . . . . 8 ⊢ (𝑅 ∈ DirRel → (𝐴𝑅𝑥 → 𝑥 ∈ ran 𝑅))
26		ssun2 4004	. . . . . . . . . . 11 ⊢ ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
27		dmrnssfld 5617	. . . . . . . . . . 11 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅
28	26, 27	sstri 3836	. . . . . . . . . 10 ⊢ ran 𝑅 ⊆ ∪ ∪ 𝑅
29	28, 3	syl5sseqr 3879	. . . . . . . . 9 ⊢ (𝑅 ∈ DirRel → ran 𝑅 ⊆ 𝑋)
30	29	sseld 3826	. . . . . . . 8 ⊢ (𝑅 ∈ DirRel → (𝑥 ∈ ran 𝑅 → 𝑥 ∈ 𝑋))
31	25, 30	syld 47	. . . . . . 7 ⊢ (𝑅 ∈ DirRel → (𝐴𝑅𝑥 → 𝑥 ∈ 𝑋))
32	31	adantrd 487	. . . . . 6 ⊢ (𝑅 ∈ DirRel → ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) → 𝑥 ∈ 𝑋))
33	32	ancrd 547	. . . . 5 ⊢ (𝑅 ∈ DirRel → ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) → (𝑥 ∈ 𝑋 ∧ (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))))
34	33	eximdv 2016	. . . 4 ⊢ (𝑅 ∈ DirRel → (∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) → ∃𝑥(𝑥 ∈ 𝑋 ∧ (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))))
35		df-rex 3123	. . . 4 ⊢ (∃𝑥 ∈ 𝑋 (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ↔ ∃𝑥(𝑥 ∈ 𝑋 ∧ (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
36	34, 35	syl6ibr 244	. . 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 1148	1 ⊢ ((𝑅 ∈ DirRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))

Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1111   = wceq 1656  ∃wex 1878   ∈ wcel 2164  ∀wral 3117  ∃wrex 3118   ∪ cun 3796   ⊆ wss 3798  ∪ cuni 4658   class class class wbr 4873   I cid 5249   × cxp 5340  ^◡ccnv 5341  dom cdm 5342  ran crn 5343   ↾ cres 5344   ∘ ccom 5346  Rel wrel 5347  DirRelcdir 17581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-dir 17583

This theorem is referenced by:  tailfb  32899