Theorem sltsex2 28008

Description: The second argument of surreal set less-than exists. (Contributed by Scott Fenton, 8-Dec-2021.)

Assertion

Ref	Expression
sltsex2	⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)

Proof of Theorem sltsex2

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

Step	Hyp	Ref	Expression
1		brslts 28006	. 2 ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)))
2		simplr 781	. 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) → 𝐵 ∈ V)
3	1, 2	sylbi 220	1 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2146  ∀wral 3081  Vcvv 3457   ⊆ wss 3906   class class class wbr 5111   No csur 27855   <s clts 27856   <<s cslts 28001

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 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

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 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-slts 28002

This theorem is used by:  ssslts1  28017  ssslts2  28018  conway  28023  cutsval  28024  sltstr  28031  sltsun1  28032  sltsun2  28033  etaslts  28037  etaslts2  28038  cutbdaybnd2lim  28041  lesrec  28043  eqcuts3  28048  madecut  28127  cofslts  28162  cofcut1  28164  cofcutr  28168  cutlt  28176  addsuniflem  28245  negsunif  28299  sltmuls1  28391  sltmuls2  28392  precsexlem11  28461