Theorem sltsss2 28182

Description: The second argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)

Assertion

Ref	Expression
sltsss2	⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No)

Proof of Theorem sltsss2

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

Step	Hyp	Ref	Expression
1		brslts 28178	. 2 ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)))
2		simpr2 1214	. 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) → 𝐵 ⊆ No)
3	1, 2	sylbi 220	1 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   ⊆ wss 3900   class class class wbr 5106  Nocsur 28027   <s clts 28028   <<s cslts 28173

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-slts 28174

This theorem is used by:  ssslts1  28189  ssslts2  28190  conway  28195  sltstr  28203  sltsun1  28204  sltsun2  28205  etaslts  28209  lesrec  28215  ltsrec  28217  eqcuts3  28220  cofslts  28334  coinitslts  28335  cofcut1  28336  cofcutr  28340  cutlt  28348  cutmax  28350  addsuniflem  28417  negsunif  28471  sltmuls1  28563  sltmuls2  28564  mulsuniflem  28565  mulsunif2lem  28585  precsexlem11  28633  renegscl  28914