Theorem cofcutrtime 28313

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 4176	. . . . . . . 8 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2		sstr 3916	. . . . . . . 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 3908	. . . . 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 28167	. . . . . . . . 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 7453	. . . . . . . . 9 ⊢ (𝐴 |s 𝐵) ∈ V
12	11	snid 4622	. . . . . . . 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 28158	. . . . . 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 5132	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 <s 𝑋)
17		leftval 28235	. . . . . . . 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 2846	. . . . . 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 3432	. . . . . 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 28266	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ No)
24		lesid 28124	. . . . 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 5106	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑥 ≤s 𝑦 ↔ 𝑥 ≤s 𝑥))
27	26	rspcev 3576	. . . 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 3154	. 2 ⊢ (((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (L‘𝑋)𝑥 ≤s 𝑦)
30		ssun2 4177	. . . . . . . 8 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
31		sstr 3916	. . . . . . . 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 3908	. . . . 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 28158	. . . . . 6 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → (𝐴 |s 𝐵) <s 𝑧)
42	35, 41	eqbrtrd 5126	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑋 <s 𝑧)
43		rightval 28236	. . . . . . . 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 2846	. . . . . 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 3432	. . . . . 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 28268	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ No)
50		lesid 28124	. . . . 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 5105	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 ≤s 𝑧 ↔ 𝑧 ≤s 𝑧))
53	52	rspcev 3576	. . . 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 3154	. 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 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  {crab 3412   ⊆ wss 3895   ∪ cun 4149  {csn 4583   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Nocsur 27997   <s clts 27998  bdaycbday 27999   ≤s cles 28101   <<s cslts 28143   |s ccuts 28145  Ocold 28209  Lcleft 28211  Rcright 28212

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-1o 8476  df-2o 8477  df-no 28000  df-lts 28001  df-bday 28002  df-les 28102  df-slts 28144  df-cuts 28146  df-made 28213  df-old 28214  df-left 28216  df-right 28217

This theorem is used by:  cofcutrtime1d  28314  cofcutrtime2d  28315