Theorem cofcutrtime 27941

Description: If 𝑋 is the cut of 𝐴 and 𝐵 and all of 𝐴 and 𝐵 are older than 𝑋, then ( L ‘𝑋) is cofinal with 𝐴 and ( R ‘𝑋) is coinitial with 𝐵. Note: we will call a cut where all of the elements of the cut are older than the cut itself a "timely" cut. Part of Theorem 4.02(12) of [Alling] p. 125. (Contributed by Scott Fenton, 27-Sep-2024.)

Assertion

Ref	Expression
cofcutrtime	⊢ (((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧))

Distinct variable groups:   𝑥,𝐴   𝑧,𝐴   𝑥,𝐵   𝑧,𝐵   𝑧,𝑤,𝑋   𝑥,𝑋,𝑦   𝑧,𝑋

Allowed substitution hints:   𝐴(𝑦,𝑤)   𝐵(𝑦,𝑤)

Proof of Theorem cofcutrtime

Step	Hyp	Ref	Expression
1		ssun1 4110	. . . . . . . 8 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2		sstr 3925	. . . . . . . 8 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋))) → 𝐴 ⊆ ( O ‘( bday ‘𝑋)))
3	1, 2	mpan 697	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) → 𝐴 ⊆ ( O ‘( bday ‘𝑋)))
4	3	3ad2ant1 1140	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → 𝐴 ⊆ ( O ‘( bday ‘𝑋)))
5	4	sselda 3917	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ( O ‘( bday ‘𝑋)))
6		simpl2 1200	. . . . . . . . 9 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝐴 <<s 𝐵)
7		cutcuts 27795	. . . . . . . . 9 ⊢ (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
8	6, 7	syl 17	. . . . . . . 8 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
9	8	simp2d 1150	. . . . . . 7 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝐴 <<s {(𝐴 |s 𝐵)})
10		simpr 486	. . . . . . 7 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
11		ovex 7393	. . . . . . . . 9 ⊢ (𝐴 |s 𝐵) ∈ V
12	11	snid 4597	. . . . . . . 8 ⊢ (𝐴 |s 𝐵) ∈ {(𝐴 |s 𝐵)}
13	12	a1i 11	. . . . . . 7 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐴 |s 𝐵) ∈ {(𝐴 |s 𝐵)})
14	9, 10, 13	sltssepcd 27786	. . . . . 6 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 <s (𝐴 |s 𝐵))
15		simpl3 1201	. . . . . 6 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑋 = (𝐴 |s 𝐵))
16	14, 15	breqtrrd 5103	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 <s 𝑋)
17		leftval 27863	. . . . . . . 8 ⊢ ( L ‘𝑋) = {𝑥 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑥 <s 𝑋}
18	17	a1i 11	. . . . . . 7 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → ( L ‘𝑋) = {𝑥 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑥 <s 𝑋})
19	18	eleq2d 2827	. . . . . 6 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ ( L ‘𝑋) ↔ 𝑥 ∈ {𝑥 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑥 <s 𝑋}))
20		rabid 3414	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑥 <s 𝑋} ↔ (𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋))
21	19, 20	bitrdi 289	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ ( L ‘𝑋) ↔ (𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋)))
22	5, 16, 21	mpbir2and 720	. . . 4 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ( L ‘𝑋))
23	22	leftnod 27894	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ No )
24		lesid 27753	. . . . 5 ⊢ (𝑥 ∈ No → 𝑥 ≤s 𝑥)
25	23, 24	syl 17	. . . 4 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤s 𝑥)
26		breq2 5079	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑥 ≤s 𝑦 ↔ 𝑥 ≤s 𝑥))
27	26	rspcev 3562	. . . 4 ⊢ ((𝑥 ∈ ( L ‘𝑋) ∧ 𝑥 ≤s 𝑥) → ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦)
28	22, 25, 27	syl2anc 591	. . 3 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦)
29	28	ralrimiva 3133	. 2 ⊢ (((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦)
30		ssun2 4111	. . . . . . . 8 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
31		sstr 3925	. . . . . . . 8 ⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋))) → 𝐵 ⊆ ( O ‘( bday ‘𝑋)))
32	30, 31	mpan 697	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) → 𝐵 ⊆ ( O ‘( bday ‘𝑋)))
33	32	3ad2ant1 1140	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → 𝐵 ⊆ ( O ‘( bday ‘𝑋)))
34	33	sselda 3917	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ ( O ‘( bday ‘𝑋)))
35		simpl3 1201	. . . . . 6 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑋 = (𝐴 |s 𝐵))
36		simpl2 1200	. . . . . . . . 9 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝐴 <<s 𝐵)
37	36, 7	syl 17	. . . . . . . 8 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
38	37	simp3d 1151	. . . . . . 7 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → {(𝐴 |s 𝐵)} <<s 𝐵)
39	12	a1i 11	. . . . . . 7 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → (𝐴 |s 𝐵) ∈ {(𝐴 |s 𝐵)})
40		simpr 486	. . . . . . 7 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
41	38, 39, 40	sltssepcd 27786	. . . . . 6 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → (𝐴 |s 𝐵) <s 𝑧)
42	35, 41	eqbrtrd 5097	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑋 <s 𝑧)
43		rightval 27864	. . . . . . . 8 ⊢ ( R ‘𝑋) = {𝑧 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑧}
44	43	a1i 11	. . . . . . 7 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → ( R ‘𝑋) = {𝑧 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑧})
45	44	eleq2d 2827	. . . . . 6 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → (𝑧 ∈ ( R ‘𝑋) ↔ 𝑧 ∈ {𝑧 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑧}))
46		rabid 3414	. . . . . 6 ⊢ (𝑧 ∈ {𝑧 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑧} ↔ (𝑧 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑋 <s 𝑧))
47	45, 46	bitrdi 289	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → (𝑧 ∈ ( R ‘𝑋) ↔ (𝑧 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑋 <s 𝑧)))
48	34, 42, 47	mpbir2and 720	. . . 4 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ ( R ‘𝑋))
49	48	rightnod 27896	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ No )
50		lesid 27753	. . . . 5 ⊢ (𝑧 ∈ No → 𝑧 ≤s 𝑧)
51	49, 50	syl 17	. . . 4 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ≤s 𝑧)
52		breq1 5078	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 ≤s 𝑧 ↔ 𝑧 ≤s 𝑧))
53	52	rspcev 3562	. . . 4 ⊢ ((𝑧 ∈ ( R ‘𝑋) ∧ 𝑧 ≤s 𝑧) → ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧)
54	48, 51, 53	syl2anc 591	. . 3 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧)
55	54	ralrimiva 3133	. 2 ⊢ (((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧)
56	29, 55	jca 517	1 ⊢ (((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∀wral 3055  ∃wrex 3065  {crab 3393   ∪ cun 3883   ⊆ wss 3885  {csn 4558   class class class wbr 5075  ‘cfv 6489  (class class class)co 7360   No csur 27625   <s clts 27626   bday cbday 27627   ≤s cles 27730   <<s cslts 27771   |s ccuts 27773   O cold 27837   L cleft 27839   R cright 27840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-1o 8399  df-2o 8400  df-no 27628  df-lts 27629  df-bday 27630  df-les 27731  df-slts 27772  df-cuts 27774  df-made 27841  df-old 27842  df-left 27844  df-right 27845

This theorem is referenced by:  cofcutrtime1d  27942  cofcutrtime2d  27943