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Theorem ssltex1 33312
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 33311 . 2 (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)))
2 simpll 766 . 2 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)) → 𝐴 ∈ V)
31, 2sylbi 220 1 (𝐴 <<s 𝐵𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wcel 2115  wral 3133  Vcvv 3480  wss 3919   class class class wbr 5052   No csur 33204   <s cslt 33205   <<s csslt 33307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-xp 5548  df-sslt 33308
This theorem is referenced by:  sssslt1  33317  sssslt2  33318  conway  33321  scutval  33322  sslttr  33325  ssltun1  33326  ssltun2  33327  etasslt  33331  slerec  33334
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