Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ssltex1 Structured version   Visualization version   GIF version

Theorem ssltex1 32232
Description: The first argument of surreal set less than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
Assertion
Ref Expression
ssltex1 (𝐴 <<s 𝐵𝐴 ∈ V)

Proof of Theorem ssltex1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brsslt 32231 . 2 (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)))
2 simpll 742 . 2 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)) → 𝐴 ∈ V)
31, 2sylbi 207 1 (𝐴 <<s 𝐵𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1070  wcel 2144  wral 3060  Vcvv 3349  wss 3721   class class class wbr 4784   No csur 32124   <s cslt 32125   <<s csslt 32227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-xp 5255  df-sslt 32228
This theorem is referenced by:  sssslt1  32237  sssslt2  32238  conway  32241  scutval  32242  sslttr  32245  ssltun1  32246  ssltun2  32247  etasslt  32251  slerec  32254
  Copyright terms: Public domain W3C validator