Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sssslt1 Structured version   Visualization version   GIF version

Theorem sssslt1 32243
Description: Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
Assertion
Ref Expression
sssslt1 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 <<s 𝐵)

Proof of Theorem sssslt1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssltex1 32238 . . . . 5 (𝐴 <<s 𝐵𝐴 ∈ V)
21adantr 466 . . . 4 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐴 ∈ V)
3 simpr 471 . . . 4 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶𝐴)
42, 3ssexd 4939 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 ∈ V)
5 ssltex2 32239 . . . 4 (𝐴 <<s 𝐵𝐵 ∈ V)
65adantr 466 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐵 ∈ V)
74, 6jca 501 . 2 ((𝐴 <<s 𝐵𝐶𝐴) → (𝐶 ∈ V ∧ 𝐵 ∈ V))
8 ssltss1 32240 . . . . 5 (𝐴 <<s 𝐵𝐴 No )
98adantr 466 . . . 4 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐴 No )
103, 9sstrd 3762 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 No )
11 ssltss2 32241 . . . 4 (𝐴 <<s 𝐵𝐵 No )
1211adantr 466 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐵 No )
13 ssltsep 32242 . . . 4 (𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
14 ssralv 3815 . . . 4 (𝐶𝐴 → (∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦 → ∀𝑥𝐶𝑦𝐵 𝑥 <s 𝑦))
1513, 14mpan9 496 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → ∀𝑥𝐶𝑦𝐵 𝑥 <s 𝑦)
1610, 12, 153jca 1122 . 2 ((𝐴 <<s 𝐵𝐶𝐴) → (𝐶 No 𝐵 No ∧ ∀𝑥𝐶𝑦𝐵 𝑥 <s 𝑦))
17 brsslt 32237 . 2 (𝐶 <<s 𝐵 ↔ ((𝐶 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 No 𝐵 No ∧ ∀𝑥𝐶𝑦𝐵 𝑥 <s 𝑦)))
187, 16, 17sylanbrc 572 1 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 <<s 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071  wcel 2145  wral 3061  Vcvv 3351  wss 3723   class class class wbr 4786   No csur 32130   <s cslt 32131   <<s csslt 32233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-sslt 32234
This theorem is referenced by:  scutun12  32254
  Copyright terms: Public domain W3C validator