Theorem ssltss1 33962

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 𝐵 → 𝐴 ⊆ No )

Proof of Theorem ssltss1

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brsslt 33959	. 2 ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)))
2		simpr1 1192	. 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) → 𝐴 ⊆ No )
3	1, 2	sylbi 216	1 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   ∈ wcel 2109  ∀wral 3065  Vcvv 3430   ⊆ wss 3891   class class class wbr 5078   No csur 33822   <s cslt 33823   <<s csslt 33954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079  df-opab 5141  df-xp 5594  df-sslt 33955

This theorem is referenced by:  sssslt1  33968  sssslt2  33969  conway  33972  scutval  33973  sslttr  33980  ssltun1  33981  ssltun2  33982  dmscut  33984  etasslt  33986  slerec  33992  sltrec  33993  ssltdisj  33994  cofsslt  34067  coinitsslt  34068  cofcut1  34069  cofcutr  34071