Theorem sltsss2 27917

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 27913	. 2 ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)))
2		simpr2 1212	. 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 )

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   ∈ wcel 2145  ∀wral 3079  Vcvv 3457   ⊆ wss 3907   class class class wbr 5105   No csur 27762   <s clts 27763   <<s cslts 27908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-slts 27909

This theorem is referenced by:  ssslts1  27924  ssslts2  27925  conway  27930  sltstr  27938  sltsun1  27939  sltsun2  27940  etaslts  27944  lesrec  27950  ltsrec  27952  eqcuts3  27955  cofslts  28069  coinitslts  28070  cofcut1  28071  cofcutr  28075  cutlt  28083  cutmax  28085  addsuniflem  28152  negsunif  28206  sltmuls1  28298  sltmuls2  28299  mulsuniflem  28300  mulsunif2lem  28320  precsexlem11  28368  renegscl  28649