Theorem cofcutrtime 27404

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 4172	. . . . . . . 8 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2		sstr 3990	. . . . . . . 8 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋))) → 𝐴 ⊆ ( O ‘( bday ‘𝑋)))
3	1, 2	mpan 689	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) → 𝐴 ⊆ ( O ‘( bday ‘𝑋)))
4	3	3ad2ant1 1134	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → 𝐴 ⊆ ( O ‘( bday ‘𝑋)))
5	4	sselda 3982	. . . . 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 27292	. . . . . . . . 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 1144	. . . . . . 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 7439	. . . . . . . . 9 ⊢ (𝐴 |s 𝐵) ∈ V
12	11	snid 4664	. . . . . . . 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 27285	. . . . . 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 5176	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 <s 𝑋)
17		leftval 27348	. . . . . . . 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 2820	. . . . . 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 3453	. . . . . 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 712	. . . 4 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ( L ‘𝑋))
23		leftssno 27365	. . . . . 6 ⊢ ( L ‘𝑋) ⊆ No
24	23, 22	sselid 3980	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ No )
25		slerflex 27256	. . . . 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 5152	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑥 ≤s 𝑦 ↔ 𝑥 ≤s 𝑥))
28	27	rspcev 3613	. . . 4 ⊢ ((𝑥 ∈ ( L ‘𝑋) ∧ 𝑥 ≤s 𝑥) → ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦)
29	22, 26, 28	syl2anc 585	. . 3 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦)
30	29	ralrimiva 3147	. 2 ⊢ (((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦)
31		ssun2 4173	. . . . . . . 8 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
32		sstr 3990	. . . . . . . 8 ⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋))) → 𝐵 ⊆ ( O ‘( bday ‘𝑋)))
33	31, 32	mpan 689	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) → 𝐵 ⊆ ( O ‘( bday ‘𝑋)))
34	33	3ad2ant1 1134	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → 𝐵 ⊆ ( O ‘( bday ‘𝑋)))
35	34	sselda 3982	. . . . 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 1145	. . . . . . 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 486	. . . . . . 7 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
42	39, 40, 41	ssltsepcd 27285	. . . . . 6 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → (𝐴 |s 𝐵) <s 𝑧)
43	36, 42	eqbrtrd 5170	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑋 <s 𝑧)
44		rightval 27349	. . . . . . . 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 2820	. . . . . 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 3453	. . . . . 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 712	. . . 4 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ ( R ‘𝑋))
50		rightssno 27366	. . . . . 6 ⊢ ( R ‘𝑋) ⊆ No
51	50, 49	sselid 3980	. . . . 5 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ No )
52		slerflex 27256	. . . . 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 5151	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 ≤s 𝑧 ↔ 𝑧 ≤s 𝑧))
55	54	rspcev 3613	. . . 4 ⊢ ((𝑧 ∈ ( R ‘𝑋) ∧ 𝑧 ≤s 𝑧) → ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧)
56	49, 53, 55	syl2anc 585	. . 3 ⊢ ((((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ 𝐵) → ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧)
57	56	ralrimiva 3147	. 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 513	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 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071  {crab 3433   ∪ cun 3946   ⊆ wss 3948  {csn 4628   class class class wbr 5148  ‘cfv 6541  (class class class)co 7406   No csur 27133   <s cslt 27134   bday cbday 27135   ≤s csle 27237   <<s csslt 27272   |s cscut 27274   O cold 27328   L cleft 27330   R cright 27331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-1o 8463  df-2o 8464  df-no 27136  df-slt 27137  df-bday 27138  df-sle 27238  df-sslt 27273  df-scut 27275  df-made 27332  df-old 27333  df-left 27335  df-right 27336

This theorem is referenced by:  cofcutrtime1d  27405  cofcutrtime2d  27406