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Theorem coinitsslt 27968
Description: If 𝐵 is coinitial with 𝐶 and 𝐴 precedes 𝐶, then 𝐴 precedes 𝐵. (Contributed by Scott Fenton, 24-Sep-2024.)
Assertion
Ref Expression
coinitsslt ((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) → 𝐴 <<s 𝐵)
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑦)

Proof of Theorem coinitsslt
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssltex1 27846 . . 3 (𝐴 <<s 𝐶𝐴 ∈ V)
213ad2ant3 1134 . 2 ((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) → 𝐴 ∈ V)
3 simp1 1135 . 2 ((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) → 𝐵 ∈ 𝒫 No )
4 ssltss1 27848 . . 3 (𝐴 <<s 𝐶𝐴 No )
543ad2ant3 1134 . 2 ((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) → 𝐴 No )
63elpwid 4614 . 2 ((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) → 𝐵 No )
7 breq2 5152 . . . . . 6 (𝑥 = 𝑏 → (𝑦 ≤s 𝑥𝑦 ≤s 𝑏))
87rexbidv 3177 . . . . 5 (𝑥 = 𝑏 → (∃𝑦𝐶 𝑦 ≤s 𝑥 ↔ ∃𝑦𝐶 𝑦 ≤s 𝑏))
9 simp12 1203 . . . . 5 (((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) ∧ 𝑎𝐴𝑏𝐵) → ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥)
10 simp3 1137 . . . . 5 (((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) ∧ 𝑎𝐴𝑏𝐵) → 𝑏𝐵)
118, 9, 10rspcdva 3623 . . . 4 (((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) ∧ 𝑎𝐴𝑏𝐵) → ∃𝑦𝐶 𝑦 ≤s 𝑏)
12 breq1 5151 . . . . 5 (𝑦 = 𝑐 → (𝑦 ≤s 𝑏𝑐 ≤s 𝑏))
1312cbvrexvw 3236 . . . 4 (∃𝑦𝐶 𝑦 ≤s 𝑏 ↔ ∃𝑐𝐶 𝑐 ≤s 𝑏)
1411, 13sylib 218 . . 3 (((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) ∧ 𝑎𝐴𝑏𝐵) → ∃𝑐𝐶 𝑐 ≤s 𝑏)
15 simpl13 1249 . . . . . 6 ((((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) ∧ 𝑎𝐴𝑏𝐵) ∧ (𝑐𝐶𝑐 ≤s 𝑏)) → 𝐴 <<s 𝐶)
1615, 4syl 17 . . . . 5 ((((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) ∧ 𝑎𝐴𝑏𝐵) ∧ (𝑐𝐶𝑐 ≤s 𝑏)) → 𝐴 No )
17 simpl2 1191 . . . . 5 ((((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) ∧ 𝑎𝐴𝑏𝐵) ∧ (𝑐𝐶𝑐 ≤s 𝑏)) → 𝑎𝐴)
1816, 17sseldd 3996 . . . 4 ((((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) ∧ 𝑎𝐴𝑏𝐵) ∧ (𝑐𝐶𝑐 ≤s 𝑏)) → 𝑎 No )
19 ssltss2 27849 . . . . . 6 (𝐴 <<s 𝐶𝐶 No )
2015, 19syl 17 . . . . 5 ((((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) ∧ 𝑎𝐴𝑏𝐵) ∧ (𝑐𝐶𝑐 ≤s 𝑏)) → 𝐶 No )
21 simprl 771 . . . . 5 ((((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) ∧ 𝑎𝐴𝑏𝐵) ∧ (𝑐𝐶𝑐 ≤s 𝑏)) → 𝑐𝐶)
2220, 21sseldd 3996 . . . 4 ((((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) ∧ 𝑎𝐴𝑏𝐵) ∧ (𝑐𝐶𝑐 ≤s 𝑏)) → 𝑐 No )
23 simpl1 1190 . . . . . 6 ((((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) ∧ 𝑎𝐴𝑏𝐵) ∧ (𝑐𝐶𝑐 ≤s 𝑏)) → (𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶))
2423, 6syl 17 . . . . 5 ((((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) ∧ 𝑎𝐴𝑏𝐵) ∧ (𝑐𝐶𝑐 ≤s 𝑏)) → 𝐵 No )
25 simpl3 1192 . . . . 5 ((((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) ∧ 𝑎𝐴𝑏𝐵) ∧ (𝑐𝐶𝑐 ≤s 𝑏)) → 𝑏𝐵)
2624, 25sseldd 3996 . . . 4 ((((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) ∧ 𝑎𝐴𝑏𝐵) ∧ (𝑐𝐶𝑐 ≤s 𝑏)) → 𝑏 No )
2715, 17, 21ssltsepcd 27854 . . . 4 ((((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) ∧ 𝑎𝐴𝑏𝐵) ∧ (𝑐𝐶𝑐 ≤s 𝑏)) → 𝑎 <s 𝑐)
28 simprr 773 . . . 4 ((((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) ∧ 𝑎𝐴𝑏𝐵) ∧ (𝑐𝐶𝑐 ≤s 𝑏)) → 𝑐 ≤s 𝑏)
2918, 22, 26, 27, 28sltletrd 27820 . . 3 ((((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) ∧ 𝑎𝐴𝑏𝐵) ∧ (𝑐𝐶𝑐 ≤s 𝑏)) → 𝑎 <s 𝑏)
3014, 29rexlimddv 3159 . 2 (((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) ∧ 𝑎𝐴𝑏𝐵) → 𝑎 <s 𝑏)
312, 3, 5, 6, 30ssltd 27851 1 ((𝐵 ∈ 𝒫 No ∧ ∀𝑥𝐵𝑦𝐶 𝑦 ≤s 𝑥𝐴 <<s 𝐶) → 𝐴 <<s 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  wss 3963  𝒫 cpw 4605   class class class wbr 5148   No csur 27699   <s cslt 27700   ≤s csle 27804   <<s csslt 27840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-sle 27805  df-sslt 27841
This theorem is referenced by:  cofcut1  27969  cofcut2  27971
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