Theorem ssltss2 27034

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 𝐵 → 𝐵 ⊆ No )

Proof of Theorem ssltss2

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

Step	Hyp	Ref	Expression
1		brsslt 27030	. 2 ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)))
2		simpr2 1195	. 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) → 𝐵 ⊆ No )
3	1, 2	sylbi 216	1 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1087   ∈ wcel 2106  ∀wral 3062  Vcvv 3442   ⊆ wss 3901   class class class wbr 5096   No csur 26893   <s cslt 26894   <<s csslt 27025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5247  ax-nul 5254  ax-pr 5376

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3444  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4274  df-if 4478  df-sn 4578  df-pr 4580  df-op 4584  df-br 5097  df-opab 5159  df-xp 5630  df-sslt 27026

This theorem is referenced by:  sssslt1  27039  sssslt2  27040  conway  27043  sslttr  27051  ssltun1  27052  ssltun2  27053  etasslt  27057  slerec  27063  sltrec  27064  cofsslt  34196  coinitsslt  34197  cofcut1  34198  cofcutr  34200  addsunif  34256