Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sssslt2 Structured version   Visualization version   GIF version

Theorem sssslt2 32822
Description: Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
Assertion
Ref Expression
sssslt2 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 <<s 𝐶)

Proof of Theorem sssslt2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssltex1 32816 . . 3 (𝐴 <<s 𝐵𝐴 ∈ V)
21adantr 473 . 2 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 ∈ V)
3 ssltex2 32817 . . . 4 (𝐴 <<s 𝐵𝐵 ∈ V)
43adantr 473 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐵 ∈ V)
5 simpr 477 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐶𝐵)
64, 5ssexd 5088 . 2 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐶 ∈ V)
7 ssltss1 32818 . . . 4 (𝐴 <<s 𝐵𝐴 No )
87adantr 473 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 No )
9 ssltss2 32819 . . . . 5 (𝐴 <<s 𝐵𝐵 No )
109adantr 473 . . . 4 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐵 No )
115, 10sstrd 3870 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐶 No )
12 ssltsep 32820 . . . . 5 (𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
1312adantr 473 . . . 4 ((𝐴 <<s 𝐵𝐶𝐵) → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
14 ssralv 3925 . . . . . 6 (𝐶𝐵 → (∀𝑦𝐵 𝑥 <s 𝑦 → ∀𝑦𝐶 𝑥 <s 𝑦))
155, 14syl 17 . . . . 5 ((𝐴 <<s 𝐵𝐶𝐵) → (∀𝑦𝐵 𝑥 <s 𝑦 → ∀𝑦𝐶 𝑥 <s 𝑦))
1615ralimdv 3130 . . . 4 ((𝐴 <<s 𝐵𝐶𝐵) → (∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦 → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦))
1713, 16mpd 15 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)
188, 11, 173jca 1109 . 2 ((𝐴 <<s 𝐵𝐶𝐵) → (𝐴 No 𝐶 No ∧ ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦))
19 brsslt 32815 . 2 (𝐴 <<s 𝐶 ↔ ((𝐴 ∈ V ∧ 𝐶 ∈ V) ∧ (𝐴 No 𝐶 No ∧ ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)))
202, 6, 18, 19syl21anbrc 1325 1 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 <<s 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1069  wcel 2051  wral 3090  Vcvv 3417  wss 3831   class class class wbr 4934   No csur 32708   <s cslt 32709   <<s csslt 32811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2752  ax-sep 5064  ax-nul 5071  ax-pr 5190
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2551  df-eu 2589  df-clab 2761  df-cleq 2773  df-clel 2848  df-nfc 2920  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3419  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4182  df-if 4354  df-sn 4445  df-pr 4447  df-op 4451  df-br 4935  df-opab 4997  df-xp 5417  df-sslt 32812
This theorem is referenced by:  scutun12  32832
  Copyright terms: Public domain W3C validator