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Theorem ssltex2 32027
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 32025 . 2 (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)))
2 simplr 807 . 2 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)) → 𝐵 ∈ V)
31, 2sylbi 207 1 (𝐴 <<s 𝐵𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054  wcel 2030  wral 2941  Vcvv 3231  wss 3607   class class class wbr 4685   No csur 31918   <s cslt 31919   <<s csslt 32021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-sslt 32022
This theorem is referenced by:  sssslt1  32031  sssslt2  32032  conway  32035  scutval  32036  sslttr  32039  ssltun1  32040  ssltun2  32041  etasslt  32045  slerec  32048
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