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Theorem ssltex1 27846
Description: The first argument of surreal set less-than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
Assertion
Ref Expression
ssltex1 (𝐴 <<s 𝐵𝐴 ∈ V)

Proof of Theorem ssltex1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brsslt 27845 . 2 (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)))
2 simpll 767 . 2 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)) → 𝐴 ∈ V)
31, 2sylbi 217 1 (𝐴 <<s 𝐵𝐴 ∈ V)
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-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:  sssslt1  27855  sssslt2  27856  conway  27859  scutval  27860  sslttr  27867  ssltun1  27868  ssltun2  27869  etasslt  27873  etasslt2  27874  scutbdaybnd2lim  27877  slerec  27879  madecut  27936  coinitsslt  27968  cofcut1  27969  cofcutr  27973  cutlt  27981  addsuniflem  28049  negsunif  28102  ssltmul1  28188  ssltmul2  28189  precsexlem11  28256
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