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Theorem cofsslt 34088
Description: If every element of 𝐴 is bounded by some element of 𝐵 and 𝐵 precedes 𝐶, then 𝐴 precedes 𝐶. Note - we will often use the term "cofinal" to denote that every element of 𝐴 is bounded above by some element of 𝐵. Similarly, we will use the term "coinitial" to denote that every element of 𝐴 is bounded below by some element of 𝐵. (Contributed by Scott Fenton, 24-Sep-2024.)
Assertion
Ref Expression
cofsslt ((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) → 𝐴 <<s 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐴(𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem cofsslt
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1135 . 2 ((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) → 𝐴 ∈ 𝒫 No )
2 ssltex2 33982 . . 3 (𝐵 <<s 𝐶𝐶 ∈ V)
323ad2ant3 1134 . 2 ((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) → 𝐶 ∈ V)
41elpwid 4544 . 2 ((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) → 𝐴 No )
5 ssltss2 33984 . . 3 (𝐵 <<s 𝐶𝐶 No )
653ad2ant3 1134 . 2 ((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) → 𝐶 No )
7 breq1 5077 . . . . . 6 (𝑥 = 𝑎 → (𝑥 ≤s 𝑦𝑎 ≤s 𝑦))
87rexbidv 3226 . . . . 5 (𝑥 = 𝑎 → (∃𝑦𝐵 𝑥 ≤s 𝑦 ↔ ∃𝑦𝐵 𝑎 ≤s 𝑦))
9 simp12 1203 . . . . 5 (((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) → ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦)
10 simp2 1136 . . . . 5 (((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) → 𝑎𝐴)
118, 9, 10rspcdva 3562 . . . 4 (((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) → ∃𝑦𝐵 𝑎 ≤s 𝑦)
12 breq2 5078 . . . . 5 (𝑦 = 𝑏 → (𝑎 ≤s 𝑦𝑎 ≤s 𝑏))
1312cbvrexvw 3384 . . . 4 (∃𝑦𝐵 𝑎 ≤s 𝑦 ↔ ∃𝑏𝐵 𝑎 ≤s 𝑏)
1411, 13sylib 217 . . 3 (((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) → ∃𝑏𝐵 𝑎 ≤s 𝑏)
15 simpl11 1247 . . . . . 6 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝐴 ∈ 𝒫 No )
1615elpwid 4544 . . . . 5 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝐴 No )
17 simpl2 1191 . . . . 5 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑎𝐴)
1816, 17sseldd 3922 . . . 4 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑎 No )
19 simpl13 1249 . . . . . 6 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝐵 <<s 𝐶)
20 ssltss1 33983 . . . . . 6 (𝐵 <<s 𝐶𝐵 No )
2119, 20syl 17 . . . . 5 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝐵 No )
22 simprl 768 . . . . 5 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑏𝐵)
2321, 22sseldd 3922 . . . 4 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑏 No )
2419, 5syl 17 . . . . 5 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝐶 No )
25 simpl3 1192 . . . . 5 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑐𝐶)
2624, 25sseldd 3922 . . . 4 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑐 No )
27 simprr 770 . . . 4 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑎 ≤s 𝑏)
2819, 22, 25ssltsepcd 33988 . . . 4 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑏 <s 𝑐)
2918, 23, 26, 27, 28slelttrd 33964 . . 3 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑎 <s 𝑐)
3014, 29rexlimddv 3220 . 2 (((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) → 𝑎 <s 𝑐)
311, 3, 4, 6, 30ssltd 33986 1 ((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) → 𝐴 <<s 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086  wcel 2106  wral 3064  wrex 3065  Vcvv 3432  wss 3887  𝒫 cpw 4533   class class class wbr 5074   No csur 33843   <s cslt 33844   ≤s csle 33947   <<s csslt 33975
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-1o 8297  df-2o 8298  df-no 33846  df-slt 33847  df-sle 33948  df-sslt 33976
This theorem is referenced by:  cofcut1  34090  cofcut2  34091
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