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Theorem ssltex1 32243
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 32242 . 2 (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)))
2 simpll 774 . 2 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)) → 𝐴 ∈ V)
31, 2sylbi 208 1 (𝐴 <<s 𝐵𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100  wcel 2157  wral 3107  Vcvv 3402  wss 3780   class class class wbr 4855   No csur 32135   <s cslt 32136   <<s csslt 32238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pr 5109
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 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-br 4856  df-opab 4918  df-xp 5330  df-sslt 32239
This theorem is referenced by:  sssslt1  32248  sssslt2  32249  conway  32252  scutval  32253  sslttr  32256  ssltun1  32257  ssltun2  32258  etasslt  32262  slerec  32265
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