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Theorem ssltex2 32872
Description: The second argument of surreal set less than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
Assertion
Ref Expression
ssltex2 (𝐴 <<s 𝐵𝐵 ∈ V)

Proof of Theorem ssltex2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brsslt 32870 . 2 (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)))
2 simplr 765 . 2 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)) → 𝐵 ∈ V)
31, 2sylbi 218 1 (𝐴 <<s 𝐵𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1080  wcel 2081  wral 3105  Vcvv 3437  wss 3863   class class class wbr 4966   No csur 32763   <s cslt 32764   <<s csslt 32866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pr 5226
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-sn 4477  df-pr 4479  df-op 4483  df-br 4967  df-opab 5029  df-xp 5454  df-sslt 32867
This theorem is referenced by:  sssslt1  32876  sssslt2  32877  conway  32880  scutval  32881  sslttr  32884  ssltun1  32885  ssltun2  32886  etasslt  32890  slerec  32893
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