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Theorem cofsslt 27825
Description: If every element of 𝐴 is bounded above 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 1134 . 2 ((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) → 𝐴 ∈ 𝒫 No )
2 ssltex2 27707 . . 3 (𝐵 <<s 𝐶𝐶 ∈ V)
323ad2ant3 1133 . 2 ((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) → 𝐶 ∈ V)
41elpwid 4607 . 2 ((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) → 𝐴 No )
5 ssltss2 27709 . . 3 (𝐵 <<s 𝐶𝐶 No )
653ad2ant3 1133 . 2 ((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) → 𝐶 No )
7 breq1 5145 . . . . . 6 (𝑥 = 𝑎 → (𝑥 ≤s 𝑦𝑎 ≤s 𝑦))
87rexbidv 3173 . . . . 5 (𝑥 = 𝑎 → (∃𝑦𝐵 𝑥 ≤s 𝑦 ↔ ∃𝑦𝐵 𝑎 ≤s 𝑦))
9 simp12 1202 . . . . 5 (((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) → ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦)
10 simp2 1135 . . . . 5 (((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) → 𝑎𝐴)
118, 9, 10rspcdva 3608 . . . 4 (((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) → ∃𝑦𝐵 𝑎 ≤s 𝑦)
12 breq2 5146 . . . . 5 (𝑦 = 𝑏 → (𝑎 ≤s 𝑦𝑎 ≤s 𝑏))
1312cbvrexvw 3230 . . . 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 1246 . . . . . 6 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝐴 ∈ 𝒫 No )
1615elpwid 4607 . . . . 5 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝐴 No )
17 simpl2 1190 . . . . 5 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑎𝐴)
1816, 17sseldd 3979 . . . 4 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑎 No )
19 simpl13 1248 . . . . . 6 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝐵 <<s 𝐶)
20 ssltss1 27708 . . . . . 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 770 . . . . 5 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑏𝐵)
2321, 22sseldd 3979 . . . 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 1191 . . . . 5 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑐𝐶)
2624, 25sseldd 3979 . . . 4 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑐 No )
27 simprr 772 . . . 4 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑎 ≤s 𝑏)
2819, 22, 25ssltsepcd 27714 . . . 4 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑏 <s 𝑐)
2918, 23, 26, 27, 28slelttrd 27681 . . 3 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑎 <s 𝑐)
3014, 29rexlimddv 3156 . 2 (((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) → 𝑎 <s 𝑐)
311, 3, 4, 6, 30ssltd 27711 1 ((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) → 𝐴 <<s 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085  wcel 2099  wral 3056  wrex 3065  Vcvv 3469  wss 3944  𝒫 cpw 4598   class class class wbr 5142   No csur 27560   <s cslt 27561   ≤s csle 27664   <<s csslt 27700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-1o 8480  df-2o 8481  df-no 27563  df-slt 27564  df-sle 27665  df-sslt 27701
This theorem is referenced by:  cofcut1  27827  cofcut2  27829
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