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Theorem sssslt2 27841
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 27831 . . 3 (𝐴 <<s 𝐵𝐴 ∈ V)
21adantr 480 . 2 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 ∈ V)
3 ssltex2 27832 . . . 4 (𝐴 <<s 𝐵𝐵 ∈ V)
43adantr 480 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐵 ∈ V)
5 simpr 484 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐶𝐵)
64, 5ssexd 5324 . 2 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐶 ∈ V)
7 ssltss1 27833 . . . 4 (𝐴 <<s 𝐵𝐴 No )
87adantr 480 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 No )
9 ssltss2 27834 . . . . 5 (𝐴 <<s 𝐵𝐵 No )
109adantr 480 . . . 4 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐵 No )
115, 10sstrd 3994 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐶 No )
12 ssltsep 27835 . . . 4 (𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
13 ssralv 4052 . . . . 5 (𝐶𝐵 → (∀𝑦𝐵 𝑥 <s 𝑦 → ∀𝑦𝐶 𝑥 <s 𝑦))
1413ralimdv 3169 . . . 4 (𝐶𝐵 → (∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦 → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦))
1512, 14mpan9 506 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)
168, 11, 153jca 1129 . 2 ((𝐴 <<s 𝐵𝐶𝐵) → (𝐴 No 𝐶 No ∧ ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦))
17 brsslt 27830 . 2 (𝐴 <<s 𝐶 ↔ ((𝐴 ∈ V ∧ 𝐶 ∈ V) ∧ (𝐴 No 𝐶 No ∧ ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)))
182, 6, 16, 17syl21anbrc 1345 1 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 <<s 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087  wcel 2108  wral 3061  Vcvv 3480  wss 3951   class class class wbr 5143   No csur 27684   <s cslt 27685   <<s csslt 27825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-sslt 27826
This theorem is referenced by:  scutun12  27855  cutmin  27969
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