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Theorem ssltss2 27151
Description: The second argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
Assertion
Ref Expression
ssltss2 (𝐴 <<s 𝐵𝐵 No )

Proof of Theorem ssltss2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brsslt 27147 . 2 (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)))
2 simpr2 1196 . 2 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)) → 𝐵 No )
31, 2sylbi 216 1 (𝐴 <<s 𝐵𝐵 No )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088  wcel 2107  wral 3061  Vcvv 3444  wss 3911   class class class wbr 5106   No csur 27004   <s cslt 27005   <<s csslt 27142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-sslt 27143
This theorem is referenced by:  sssslt1  27156  sssslt2  27157  conway  27160  sslttr  27168  ssltun1  27169  ssltun2  27170  etasslt  27174  slerec  27180  sltrec  27181  cofsslt  27259  coinitsslt  27260  cofcut1  27261  cofcutr  27265  addsunif  27332  negsunif  27372
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