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Theorem cofsslt 33743
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 1137 . 2 ((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) → 𝐴 ∈ 𝒫 No )
2 ssltex2 33637 . . 3 (𝐵 <<s 𝐶𝐶 ∈ V)
323ad2ant3 1136 . 2 ((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) → 𝐶 ∈ V)
41elpwid 4509 . 2 ((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) → 𝐴 No )
5 ssltss2 33639 . . 3 (𝐵 <<s 𝐶𝐶 No )
653ad2ant3 1136 . 2 ((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) → 𝐶 No )
7 breq1 5043 . . . . . 6 (𝑥 = 𝑎 → (𝑥 ≤s 𝑦𝑎 ≤s 𝑦))
87rexbidv 3208 . . . . 5 (𝑥 = 𝑎 → (∃𝑦𝐵 𝑥 ≤s 𝑦 ↔ ∃𝑦𝐵 𝑎 ≤s 𝑦))
9 simp12 1205 . . . . 5 (((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) → ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦)
10 simp2 1138 . . . . 5 (((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) → 𝑎𝐴)
118, 9, 10rspcdva 3531 . . . 4 (((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) → ∃𝑦𝐵 𝑎 ≤s 𝑦)
12 breq2 5044 . . . . 5 (𝑦 = 𝑏 → (𝑎 ≤s 𝑦𝑎 ≤s 𝑏))
1312cbvrexvw 3351 . . . 4 (∃𝑦𝐵 𝑎 ≤s 𝑦 ↔ ∃𝑏𝐵 𝑎 ≤s 𝑏)
1411, 13sylib 221 . . 3 (((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) → ∃𝑏𝐵 𝑎 ≤s 𝑏)
15 simpl11 1249 . . . . . 6 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝐴 ∈ 𝒫 No )
1615elpwid 4509 . . . . 5 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝐴 No )
17 simpl2 1193 . . . . 5 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑎𝐴)
1816, 17sseldd 3888 . . . 4 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑎 No )
19 simpl13 1251 . . . . . 6 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝐵 <<s 𝐶)
20 ssltss1 33638 . . . . . 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 771 . . . . 5 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑏𝐵)
2321, 22sseldd 3888 . . . 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 1194 . . . . 5 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑐𝐶)
2624, 25sseldd 3888 . . . 4 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑐 No )
27 simprr 773 . . . 4 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑎 ≤s 𝑏)
2819, 22, 25ssltsepcd 33643 . . . 4 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑏 <s 𝑐)
2918, 23, 26, 27, 28slelttrd 33619 . . 3 ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) ∧ (𝑏𝐵𝑎 ≤s 𝑏)) → 𝑎 <s 𝑐)
3014, 29rexlimddv 3202 . 2 (((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) ∧ 𝑎𝐴𝑐𝐶) → 𝑎 <s 𝑐)
311, 3, 4, 6, 30ssltd 33641 1 ((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ≤s 𝑦𝐵 <<s 𝐶) → 𝐴 <<s 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088  wcel 2114  wral 3054  wrex 3055  Vcvv 3400  wss 3853  𝒫 cpw 4498   class class class wbr 5040   No csur 33498   <s cslt 33499   ≤s csle 33602   <<s csslt 33630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-ord 6185  df-on 6186  df-suc 6188  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-1o 8143  df-2o 8144  df-no 33501  df-slt 33502  df-sle 33603  df-sslt 33631
This theorem is referenced by:  cofcut1  33745  cofcut2  33746
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