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Theorem ssltss1 32221
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 32218 . 2 (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)))
2 simpr1 1241 . 2 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)) → 𝐴 No )
31, 2sylbi 208 1 (𝐴 <<s 𝐵𝐴 No )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100  wcel 2158  wral 3095  Vcvv 3390  wss 3766   class class class wbr 4840   No csur 32111   <s cslt 32112   <<s csslt 32214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-sep 4971  ax-nul 4980  ax-pr 5093
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ral 3100  df-rex 3101  df-rab 3104  df-v 3392  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-nul 4114  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-br 4841  df-opab 4903  df-xp 5314  df-sslt 32215
This theorem is referenced by:  sssslt1  32224  sssslt2  32225  conway  32228  scutval  32229  sslttr  32232  ssltun1  32233  ssltun2  32234  dmscut  32236  etasslt  32238  slerec  32241  sltrec  32242
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