Theorem cofcutrtime 27835

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 4141	. . . . . . . 8 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2		sstr 3955	. . . . . . . 8 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋))) → 𝐴 ⊆ ( O ‘( bday ‘𝑋)))
3	1, 2	mpan 690	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) → 𝐴 ⊆ ( O ‘( bday ‘𝑋)))
4	3	3ad2ant1 1133	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → 𝐴 ⊆ ( O ‘( bday ‘𝑋)))
5	4	sselda 3946	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ( O ‘( bday ‘𝑋)))
6		simpl2 1193	. . . . . . . . 9 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝐴 <<s 𝐵)
7		scutcut 27713	. . . . . . . . 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 1143	. . . . . . 7 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝐴 <<s {(𝐴 |s 𝐵)})
10		simpr 484	. . . . . . 7 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
11		ovex 7420	. . . . . . . . 9 ⊢ (𝐴 |s 𝐵) ∈ V
12	11	snid 4626	. . . . . . . 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	ssltsepcd 27706	. . . . . 6 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 <s (𝐴 |s 𝐵))
15		simpl3 1194	. . . . . 6 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑋 = (𝐴 |s 𝐵))
16	14, 15	breqtrrd 5135	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 <s 𝑋)
17		leftval 27771	. . . . . . . 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 2814	. . . . . 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 3427	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑥 <s 𝑋} ↔ (𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋))
21	19, 20	bitrdi 287	. . . . 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 713	. . . 4 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ( L ‘𝑋))
23		leftssno 27792	. . . . . 6 ⊢ ( L ‘𝑋) ⊆ No
24	23, 22	sselid 3944	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ No )
25		slerflex 27675	. . . . 5 ⊢ (𝑥 ∈ No → 𝑥 ≤s 𝑥)
26	24, 25	syl 17	. . . 4 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤s 𝑥)
27		breq2 5111	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑥 ≤s 𝑦 ↔ 𝑥 ≤s 𝑥))
28	27	rspcev 3588	. . . 4 ⊢ ((𝑥 ∈ ( L ‘𝑋) ∧ 𝑥 ≤s 𝑥) → ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦)
29	22, 26, 28	syl2anc 584	. . 3 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦)
30	29	ralrimiva 3125	. 2 ⊢ (((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦)
31		ssun2 4142	. . . . . . . 8 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
32		sstr 3955	. . . . . . . 8 ⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋))) → 𝐵 ⊆ ( O ‘( bday ‘𝑋)))
33	31, 32	mpan 690	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) → 𝐵 ⊆ ( O ‘( bday ‘𝑋)))
34	33	3ad2ant1 1133	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → 𝐵 ⊆ ( O ‘( bday ‘𝑋)))
35	34	sselda 3946	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ ( O ‘( bday ‘𝑋)))
36		simpl3 1194	. . . . . 6 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑋 = (𝐴 |s 𝐵))
37		simpl2 1193	. . . . . . . . 9 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝐴 <<s 𝐵)
38	37, 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))
39	38	simp3d 1144	. . . . . . 7 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → {(𝐴 |s 𝐵)} <<s 𝐵)
40	12	a1i 11	. . . . . . 7 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → (𝐴 |s 𝐵) ∈ {(𝐴 |s 𝐵)})
41		simpr 484	. . . . . . 7 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
42	39, 40, 41	ssltsepcd 27706	. . . . . 6 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → (𝐴 |s 𝐵) <s 𝑧)
43	36, 42	eqbrtrd 5129	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑋 <s 𝑧)
44		rightval 27772	. . . . . . . 8 ⊢ ( R ‘𝑋) = {𝑧 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑧}
45	44	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})
46	45	eleq2d 2814	. . . . . 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}))
47		rabid 3427	. . . . . 6 ⊢ (𝑧 ∈ {𝑧 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑧} ↔ (𝑧 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑋 <s 𝑧))
48	46, 47	bitrdi 287	. . . . 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)))
49	35, 43, 48	mpbir2and 713	. . . 4 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ ( R ‘𝑋))
50		rightssno 27793	. . . . . 6 ⊢ ( R ‘𝑋) ⊆ No
51	50, 49	sselid 3944	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ No )
52		slerflex 27675	. . . . 5 ⊢ (𝑧 ∈ No → 𝑧 ≤s 𝑧)
53	51, 52	syl 17	. . . 4 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ≤s 𝑧)
54		breq1 5110	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 ≤s 𝑧 ↔ 𝑧 ≤s 𝑧))
55	54	rspcev 3588	. . . 4 ⊢ ((𝑧 ∈ ( R ‘𝑋) ∧ 𝑧 ≤s 𝑧) → ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧)
56	49, 53, 55	syl2anc 584	. . 3 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧)
57	56	ralrimiva 3125	. 2 ⊢ (((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧)
58	30, 57	jca 511	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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3405   ∪ cun 3912   ⊆ wss 3914  {csn 4589   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387   No csur 27551   <s cslt 27552   bday cbday 27553   ≤s csle 27656   <<s csslt 27692   |s cscut 27694   O cold 27751   L cleft 27753   R cright 27754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-1o 8434  df-2o 8435  df-no 27554  df-slt 27555  df-bday 27556  df-sle 27657  df-sslt 27693  df-scut 27695  df-made 27755  df-old 27756  df-left 27758  df-right 27759

This theorem is referenced by:  cofcutrtime1d  27836  cofcutrtime2d  27837