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Theorem ssltex2 27079
Description: The second argument of surreal set less-than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
Assertion
Ref Expression
ssltex2 (𝐴 <<s 𝐵𝐵 ∈ V)

Proof of Theorem ssltex2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brsslt 27077 . 2 (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)))
2 simplr 768 . 2 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)) → 𝐵 ∈ V)
31, 2sylbi 216 1 (𝐴 <<s 𝐵𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088  wcel 2107  wral 3063  Vcvv 3444  wss 3909   class class class wbr 5104   No csur 26940   <s cslt 26941   <<s csslt 27072
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 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
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 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-xp 5638  df-sslt 27073
This theorem is referenced by:  sssslt1  27086  sssslt2  27087  conway  27090  scutval  27091  sslttr  27098  ssltun1  27099  ssltun2  27100  etasslt  27104  etasslt2  27105  scutbdaybnd2lim  27108  slerec  27110  madecut  27163  cofsslt  27186  cofcut1  27188  cofcutr  27192  addsunif  34301
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