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Theorem sssslt2 27856
Description: Relation between surreal set less-than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
Assertion
Ref Expression
sssslt2 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 <<s 𝐶)

Proof of Theorem sssslt2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssltex1 27846 . . 3 (𝐴 <<s 𝐵𝐴 ∈ V)
21adantr 480 . 2 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 ∈ V)
3 ssltex2 27847 . . . 4 (𝐴 <<s 𝐵𝐵 ∈ V)
43adantr 480 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐵 ∈ V)
5 simpr 484 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐶𝐵)
64, 5ssexd 5330 . 2 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐶 ∈ V)
7 ssltss1 27848 . . . 4 (𝐴 <<s 𝐵𝐴 No )
87adantr 480 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 No )
9 ssltss2 27849 . . . . 5 (𝐴 <<s 𝐵𝐵 No )
109adantr 480 . . . 4 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐵 No )
115, 10sstrd 4006 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐶 No )
12 ssltsep 27850 . . . 4 (𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
13 ssralv 4064 . . . . 5 (𝐶𝐵 → (∀𝑦𝐵 𝑥 <s 𝑦 → ∀𝑦𝐶 𝑥 <s 𝑦))
1413ralimdv 3167 . . . 4 (𝐶𝐵 → (∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦 → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦))
1512, 14mpan9 506 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)
168, 11, 153jca 1127 . 2 ((𝐴 <<s 𝐵𝐶𝐵) → (𝐴 No 𝐶 No ∧ ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦))
17 brsslt 27845 . 2 (𝐴 <<s 𝐶 ↔ ((𝐴 ∈ V ∧ 𝐶 ∈ V) ∧ (𝐴 No 𝐶 No ∧ ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)))
182, 6, 16, 17syl21anbrc 1343 1 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 <<s 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086  wcel 2106  wral 3059  Vcvv 3478  wss 3963   class class class wbr 5148   No csur 27699   <s cslt 27700   <<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-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-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-sslt 27841
This theorem is referenced by:  scutun12  27870  cutmin  27984
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