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Theorem ssltss1 33144
Description: The first argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
Assertion
Ref Expression
ssltss1 (𝐴 <<s 𝐵𝐴 No )

Proof of Theorem ssltss1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brsslt 33141 . 2 (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)))
2 simpr1 1188 . 2 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)) → 𝐴 No )
31, 2sylbi 218 1 (𝐴 <<s 𝐵𝐴 No )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081  wcel 2107  wral 3142  Vcvv 3499  wss 3939   class class class wbr 5062   No csur 33034   <s cslt 33035   <<s csslt 33137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-xp 5559  df-sslt 33138
This theorem is referenced by:  sssslt1  33147  sssslt2  33148  conway  33151  scutval  33152  sslttr  33155  ssltun1  33156  ssltun2  33157  dmscut  33159  etasslt  33161  slerec  33164  sltrec  33165
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