Theorem sltsex2 28159

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 28157	. 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 2144  ∀wral 3076  Vcvv 3450   ⊆ wss 3896   class class class wbr 5102  Nocsur 28006   <s clts 28007   <<s cslts 28152

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 5391

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 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5657  df-slts 28153

This theorem is used by:  ssslts1  28168  ssslts2  28169  conway  28174  cutsval  28175  sltstr  28182  sltsun1  28183  sltsun2  28184  etaslts  28188  etaslts2  28189  cutbdaybnd2lim  28192  lesrec  28194  eqcuts3  28199  madecut  28278  cofslts  28313  cofcut1  28315  cofcutr  28319  cutlt  28327  addsuniflem  28396  negsunif  28450  sltmuls1  28542  sltmuls2  28543  precsexlem11  28612