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Theorem cofcutrtime 27961
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 (((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → (∀𝑥𝐴𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤 ∈ ( 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
StepHypRef Expression
1 ssun1 4178 . . . . . . . 8 𝐴 ⊆ (𝐴𝐵)
2 sstr 3992 . . . . . . . 8 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ⊆ ( O ‘( bday 𝑋))) → 𝐴 ⊆ ( O ‘( bday 𝑋)))
31, 2mpan 690 . . . . . . 7 ((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) → 𝐴 ⊆ ( O ‘( bday 𝑋)))
433ad2ant1 1134 . . . . . 6 (((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → 𝐴 ⊆ ( O ‘( bday 𝑋)))
54sselda 3983 . . . . 5 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → 𝑥 ∈ ( O ‘( bday 𝑋)))
6 simpl2 1193 . . . . . . . . 9 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → 𝐴 <<s 𝐵)
7 scutcut 27846 . . . . . . . . 9 (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
86, 7syl 17 . . . . . . . 8 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
98simp2d 1144 . . . . . . 7 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → 𝐴 <<s {(𝐴 |s 𝐵)})
10 simpr 484 . . . . . . 7 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → 𝑥𝐴)
11 ovex 7464 . . . . . . . . 9 (𝐴 |s 𝐵) ∈ V
1211snid 4662 . . . . . . . 8 (𝐴 |s 𝐵) ∈ {(𝐴 |s 𝐵)}
1312a1i 11 . . . . . . 7 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → (𝐴 |s 𝐵) ∈ {(𝐴 |s 𝐵)})
149, 10, 13ssltsepcd 27839 . . . . . 6 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → 𝑥 <s (𝐴 |s 𝐵))
15 simpl3 1194 . . . . . 6 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → 𝑋 = (𝐴 |s 𝐵))
1614, 15breqtrrd 5171 . . . . 5 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → 𝑥 <s 𝑋)
17 leftval 27902 . . . . . . . 8 ( L ‘𝑋) = {𝑥 ∈ ( O ‘( bday 𝑋)) ∣ 𝑥 <s 𝑋}
1817a1i 11 . . . . . . 7 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → ( L ‘𝑋) = {𝑥 ∈ ( O ‘( bday 𝑋)) ∣ 𝑥 <s 𝑋})
1918eleq2d 2827 . . . . . 6 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → (𝑥 ∈ ( L ‘𝑋) ↔ 𝑥 ∈ {𝑥 ∈ ( O ‘( bday 𝑋)) ∣ 𝑥 <s 𝑋}))
20 rabid 3458 . . . . . 6 (𝑥 ∈ {𝑥 ∈ ( O ‘( bday 𝑋)) ∣ 𝑥 <s 𝑋} ↔ (𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋))
2119, 20bitrdi 287 . . . . 5 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → (𝑥 ∈ ( L ‘𝑋) ↔ (𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋)))
225, 16, 21mpbir2and 713 . . . 4 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → 𝑥 ∈ ( L ‘𝑋))
23 leftssno 27919 . . . . . 6 ( L ‘𝑋) ⊆ No
2423, 22sselid 3981 . . . . 5 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → 𝑥 No )
25 slerflex 27808 . . . . 5 (𝑥 No 𝑥 ≤s 𝑥)
2624, 25syl 17 . . . 4 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → 𝑥 ≤s 𝑥)
27 breq2 5147 . . . . 5 (𝑦 = 𝑥 → (𝑥 ≤s 𝑦𝑥 ≤s 𝑥))
2827rspcev 3622 . . . 4 ((𝑥 ∈ ( L ‘𝑋) ∧ 𝑥 ≤s 𝑥) → ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦)
2922, 26, 28syl2anc 584 . . 3 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦)
3029ralrimiva 3146 . 2 (((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → ∀𝑥𝐴𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦)
31 ssun2 4179 . . . . . . . 8 𝐵 ⊆ (𝐴𝐵)
32 sstr 3992 . . . . . . . 8 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ⊆ ( O ‘( bday 𝑋))) → 𝐵 ⊆ ( O ‘( bday 𝑋)))
3331, 32mpan 690 . . . . . . 7 ((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) → 𝐵 ⊆ ( O ‘( bday 𝑋)))
34333ad2ant1 1134 . . . . . 6 (((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → 𝐵 ⊆ ( O ‘( bday 𝑋)))
3534sselda 3983 . . . . 5 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → 𝑧 ∈ ( O ‘( bday 𝑋)))
36 simpl3 1194 . . . . . 6 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → 𝑋 = (𝐴 |s 𝐵))
37 simpl2 1193 . . . . . . . . 9 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → 𝐴 <<s 𝐵)
3837, 7syl 17 . . . . . . . 8 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
3938simp3d 1145 . . . . . . 7 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → {(𝐴 |s 𝐵)} <<s 𝐵)
4012a1i 11 . . . . . . 7 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → (𝐴 |s 𝐵) ∈ {(𝐴 |s 𝐵)})
41 simpr 484 . . . . . . 7 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → 𝑧𝐵)
4239, 40, 41ssltsepcd 27839 . . . . . 6 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → (𝐴 |s 𝐵) <s 𝑧)
4336, 42eqbrtrd 5165 . . . . 5 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → 𝑋 <s 𝑧)
44 rightval 27903 . . . . . . . 8 ( R ‘𝑋) = {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧}
4544a1i 11 . . . . . . 7 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → ( R ‘𝑋) = {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧})
4645eleq2d 2827 . . . . . 6 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → (𝑧 ∈ ( R ‘𝑋) ↔ 𝑧 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧}))
47 rabid 3458 . . . . . 6 (𝑧 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧} ↔ (𝑧 ∈ ( O ‘( bday 𝑋)) ∧ 𝑋 <s 𝑧))
4846, 47bitrdi 287 . . . . 5 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → (𝑧 ∈ ( R ‘𝑋) ↔ (𝑧 ∈ ( O ‘( bday 𝑋)) ∧ 𝑋 <s 𝑧)))
4935, 43, 48mpbir2and 713 . . . 4 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → 𝑧 ∈ ( R ‘𝑋))
50 rightssno 27920 . . . . . 6 ( R ‘𝑋) ⊆ No
5150, 49sselid 3981 . . . . 5 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → 𝑧 No )
52 slerflex 27808 . . . . 5 (𝑧 No 𝑧 ≤s 𝑧)
5351, 52syl 17 . . . 4 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → 𝑧 ≤s 𝑧)
54 breq1 5146 . . . . 5 (𝑤 = 𝑧 → (𝑤 ≤s 𝑧𝑧 ≤s 𝑧))
5554rspcev 3622 . . . 4 ((𝑧 ∈ ( R ‘𝑋) ∧ 𝑧 ≤s 𝑧) → ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧)
5649, 53, 55syl2anc 584 . . 3 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧)
5756ralrimiva 3146 . 2 (((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → ∀𝑧𝐵𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧)
5830, 57jca 511 1 (((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → (∀𝑥𝐴𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  {crab 3436  cun 3949  wss 3951  {csn 4626   class class class wbr 5143  cfv 6561  (class class class)co 7431   No csur 27684   <s cslt 27685   bday cbday 27686   ≤s csle 27789   <<s csslt 27825   |s cscut 27827   O cold 27882   L cleft 27884   R cright 27885
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-made 27886  df-old 27887  df-left 27889  df-right 27890
This theorem is referenced by:  cofcutrtime1d  27962  cofcutrtime2d  27963
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