Theorem ssltss1 33983

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 33980	. 2 ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)))
2		simpr1 1193	. 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 396   ∧ w3a 1086   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ⊆ wss 3887   class class class wbr 5074   No csur 33843   <s cslt 33844   <<s csslt 33975

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-sslt 33976

This theorem is referenced by:  sssslt1  33989  sssslt2  33990  conway  33993  scutval  33994  sslttr  34001  ssltun1  34002  ssltun2  34003  dmscut  34005  etasslt  34007  slerec  34013  sltrec  34014  ssltdisj  34015  cofsslt  34088  coinitsslt  34089  cofcut1  34090  cofcutr  34092