Theorem sltsss1 28133

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 28130	. 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 2145  ∀wral 3077  Vcvv 3451   ⊆ wss 3899   class class class wbr 5103   No csur 27979   <s clts 27980   <<s cslts 28125

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-slts 28126

This theorem is used by:  ssslts1  28141  ssslts2  28142  conway  28147  cutsval  28148  sltstr  28155  sltsun1  28156  sltsun2  28157  dmcuts  28159  etaslts  28161  lesrec  28167  ltsrec  28169  sltsdisj  28171  eqcuts3  28172  cofslts  28286  coinitslts  28287  cofcut1  28288  cofcutr  28292  cutlt  28300  cutmin  28303  addsuniflem  28369  negsunif  28423  sltmuls1  28515  sltmuls2  28516  mulsuniflem  28517  mulsunif2lem  28537  precsexlem11  28585  renegscl  28866