Theorem cofcutrtime 28340

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 4179	. . . . . . . 8 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2		sstr 3921	. . . . . . . 8 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋))) → 𝐴 ⊆ (O‘(bday‘𝑋)))
3	1, 2	mpan 703	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) → 𝐴 ⊆ (O‘(bday‘𝑋)))
4	3	3ad2ant1 1151	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → 𝐴 ⊆ (O‘(bday‘𝑋)))
5	4	sselda 3913	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (O‘(bday‘𝑋)))
6		simpl2 1211	. . . . . . . . 9 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝐴 <<s 𝐵)
7		cutcuts 28194	. . . . . . . . 9 ⊢ (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
8	6, 7	syl 18	. . . . . . . 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 1161	. . . . . . 7 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝐴 <<s {(𝐴 |s 𝐵)})
10		simpr 490	. . . . . . 7 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
11		ovex 7473	. . . . . . . . 9 ⊢ (𝐴 |s 𝐵) ∈ V
12	11	snid 4623	. . . . . . . 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 28185	. . . . . 6 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 <s (𝐴 |s 𝐵))
15		simpl3 1212	. . . . . 6 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑋 = (𝐴 |s 𝐵))
16	14, 15	breqtrrd 5136	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 <s 𝑋)
17		leftval 28262	. . . . . . . 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 2850	. . . . . 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 3436	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ (O‘(bday‘𝑋)) ∣ 𝑥 <s 𝑋} ↔ (𝑥 ∈ (O‘(bday‘𝑋)) ∧ 𝑥 <s 𝑋))
21	19, 20	bitrdi 290	. . . . 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 726	. . . 4 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (L‘𝑋))
23	22	leftnod 28293	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ No)
24		lesid 28151	. . . . 5 ⊢ (𝑥 ∈ No → 𝑥 ≤s 𝑥)
25	23, 24	syl 18	. . . 4 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤s 𝑥)
26		breq2 5110	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑥 ≤s 𝑦 ↔ 𝑥 ≤s 𝑥))
27	26	rspcev 3580	. . . 4 ⊢ ((𝑥 ∈ (L‘𝑋) ∧ 𝑥 ≤s 𝑥) → ∃𝑦 ∈ (L‘𝑋)𝑥 ≤s 𝑦)
28	22, 25, 27	syl2anc 596	. . 3 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ (L‘𝑋)𝑥 ≤s 𝑦)
29	28	ralrimiva 3158	. 2 ⊢ (((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (L‘𝑋)𝑥 ≤s 𝑦)
30		ssun2 4180	. . . . . . . 8 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
31		sstr 3921	. . . . . . . 8 ⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋))) → 𝐵 ⊆ (O‘(bday‘𝑋)))
32	30, 31	mpan 703	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) → 𝐵 ⊆ (O‘(bday‘𝑋)))
33	32	3ad2ant1 1151	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → 𝐵 ⊆ (O‘(bday‘𝑋)))
34	33	sselda 3913	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (O‘(bday‘𝑋)))
35		simpl3 1212	. . . . . 6 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑋 = (𝐴 |s 𝐵))
36		simpl2 1211	. . . . . . . . 9 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝐴 <<s 𝐵)
37	36, 7	syl 18	. . . . . . . 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 1162	. . . . . . 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 490	. . . . . . 7 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
41	38, 39, 40	sltssepcd 28185	. . . . . 6 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → (𝐴 |s 𝐵) <s 𝑧)
42	35, 41	eqbrtrd 5130	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑋 <s 𝑧)
43		rightval 28263	. . . . . . . 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 2850	. . . . . 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 3436	. . . . . 6 ⊢ (𝑧 ∈ {𝑧 ∈ (O‘(bday‘𝑋)) ∣ 𝑋 <s 𝑧} ↔ (𝑧 ∈ (O‘(bday‘𝑋)) ∧ 𝑋 <s 𝑧))
47	45, 46	bitrdi 290	. . . . 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 726	. . . 4 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (R‘𝑋))
49	48	rightnod 28295	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ No)
50		lesid 28151	. . . . 5 ⊢ (𝑧 ∈ No → 𝑧 ≤s 𝑧)
51	49, 50	syl 18	. . . 4 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ≤s 𝑧)
52		breq1 5109	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 ≤s 𝑧 ↔ 𝑧 ≤s 𝑧))
53	52	rspcev 3580	. . . 4 ⊢ ((𝑧 ∈ (R‘𝑋) ∧ 𝑧 ≤s 𝑧) → ∃𝑤 ∈ (R‘𝑋)𝑤 ≤s 𝑧)
54	48, 51, 53	syl2anc 596	. . 3 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → ∃𝑤 ∈ (R‘𝑋)𝑤 ≤s 𝑧)
55	54	ralrimiva 3158	. 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 521	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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  {crab 3416   ⊆ wss 3900   ∪ cun 4152  {csn 4588   class class class wbr 5106  ‘cfv 6555  (class class class)co 7440  Nocsur 28024   <s clts 28025  bdaycbday 28026   ≤s cles 28128   <<s cslts 28170   |s ccuts 28172  Ocold 28236  Lcleft 28238  Rcright 28239

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-1o 8501  df-2o 8502  df-no 28027  df-lts 28028  df-bday 28029  df-les 28129  df-slts 28171  df-cuts 28173  df-made 28240  df-old 28241  df-left 28243  df-right 28244

This theorem is used by:  cofcutrtime1d  28341  cofcutrtime2d  28342