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Theorem sssslt1 27858
Description: Relation between surreal set less-than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
Assertion
Ref Expression
sssslt1 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 <<s 𝐵)

Proof of Theorem sssslt1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssltex1 27849 . . . 4 (𝐴 <<s 𝐵𝐴 ∈ V)
21adantr 480 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐴 ∈ V)
3 simpr 484 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶𝐴)
42, 3ssexd 5342 . 2 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 ∈ V)
5 ssltex2 27850 . . 3 (𝐴 <<s 𝐵𝐵 ∈ V)
65adantr 480 . 2 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐵 ∈ V)
7 ssltss1 27851 . . . . 5 (𝐴 <<s 𝐵𝐴 No )
87adantr 480 . . . 4 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐴 No )
93, 8sstrd 4019 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 No )
10 ssltss2 27852 . . . 4 (𝐴 <<s 𝐵𝐵 No )
1110adantr 480 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐵 No )
12 ssltsep 27853 . . . 4 (𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
13 ssralv 4077 . . . 4 (𝐶𝐴 → (∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦 → ∀𝑥𝐶𝑦𝐵 𝑥 <s 𝑦))
1412, 13mpan9 506 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → ∀𝑥𝐶𝑦𝐵 𝑥 <s 𝑦)
159, 11, 143jca 1128 . 2 ((𝐴 <<s 𝐵𝐶𝐴) → (𝐶 No 𝐵 No ∧ ∀𝑥𝐶𝑦𝐵 𝑥 <s 𝑦))
16 brsslt 27848 . 2 (𝐶 <<s 𝐵 ↔ ((𝐶 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 No 𝐵 No ∧ ∀𝑥𝐶𝑦𝐵 𝑥 <s 𝑦)))
174, 6, 15, 16syl21anbrc 1344 1 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 <<s 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087  wcel 2108  wral 3067  Vcvv 3488  wss 3976   class class class wbr 5166   No csur 27702   <s cslt 27703   <<s csslt 27843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-sslt 27844
This theorem is referenced by:  scutun12  27873  cutmax  27986  precsexlem11  28259
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