Theorem sltsss1 28156

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

Assertion

Ref	Expression
sltsss1	⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No)

Proof of Theorem sltsss1

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

Step	Hyp	Ref	Expression
1		brslts 28153	. 2 ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)))
2		simpr1 1213	. 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) → 𝐴 ⊆ No)
3	1, 2	sylbi 220	1 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ⊆ wss 3895   class class class wbr 5102  Nocsur 28002   <s clts 28003   <<s cslts 28148

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-slts 28149

This theorem is used by:  ssslts1  28164  ssslts2  28165  conway  28170  cutsval  28171  sltstr  28178  sltsun1  28179  sltsun2  28180  dmcuts  28182  etaslts  28184  lesrec  28190  ltsrec  28192  sltsdisj  28194  eqcuts3  28195  cofslts  28309  coinitslts  28310  cofcut1  28311  cofcutr  28315  cutlt  28323  cutmin  28326  addsuniflem  28392  negsunif  28446  sltmuls1  28538  sltmuls2  28539  mulsuniflem  28540  mulsunif2lem  28560  precsexlem11  28608  renegscl  28889