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Theorem ssltex1 27831
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 27830 . 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 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-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:  sssslt1  27840  sssslt2  27841  conway  27844  scutval  27845  sslttr  27852  ssltun1  27853  ssltun2  27854  etasslt  27858  etasslt2  27859  scutbdaybnd2lim  27862  slerec  27864  madecut  27921  coinitsslt  27953  cofcut1  27954  cofcutr  27958  cutlt  27966  addsuniflem  28034  negsunif  28087  ssltmul1  28173  ssltmul2  28174  precsexlem11  28241
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