Theorem ssltss2 33260

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 33256	. 2 ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)))
2		simpr2 1191	. 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) → 𝐵 ⊆ No )
3	1, 2	sylbi 219	1 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   ∈ wcel 2114  ∀wral 3133  Vcvv 3481   ⊆ wss 3919   class class class wbr 5047   No csur 33149   <s cslt 33150   <<s csslt 33252

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5184  ax-nul 5191  ax-pr 5311

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3483  df-dif 3922  df-un 3924  df-in 3926  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-br 5048  df-opab 5110  df-xp 5542  df-sslt 33253

This theorem is referenced by:  sssslt1  33262  sssslt2  33263  conway  33266  sslttr  33270  ssltun1  33271  ssltun2  33272  etasslt  33276  slerec  33279  sltrec  33280