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

 Description: Binary relation form of the surreal set less-than relation. (Contributed by Scott Fenton, 8-Dec-2021.)
Assertion
Ref Expression
brsslt (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sslt 33364 . . 3 <<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 No 𝑏 No ∧ ∀𝑥𝑎𝑦𝑏 𝑥 <s 𝑦)}
21bropaex12 5606 . 2 (𝐴 <<s 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 sseq1 3940 . . . 4 (𝑎 = 𝐴 → (𝑎 No 𝐴 No ))
4 raleq 3358 . . . 4 (𝑎 = 𝐴 → (∀𝑥𝑎𝑦𝑏 𝑥 <s 𝑦 ↔ ∀𝑥𝐴𝑦𝑏 𝑥 <s 𝑦))
53, 43anbi13d 1435 . . 3 (𝑎 = 𝐴 → ((𝑎 No 𝑏 No ∧ ∀𝑥𝑎𝑦𝑏 𝑥 <s 𝑦) ↔ (𝐴 No 𝑏 No ∧ ∀𝑥𝐴𝑦𝑏 𝑥 <s 𝑦)))
6 sseq1 3940 . . . 4 (𝑏 = 𝐵 → (𝑏 No 𝐵 No ))
7 raleq 3358 . . . . 5 (𝑏 = 𝐵 → (∀𝑦𝑏 𝑥 <s 𝑦 ↔ ∀𝑦𝐵 𝑥 <s 𝑦))
87ralbidv 3162 . . . 4 (𝑏 = 𝐵 → (∀𝑥𝐴𝑦𝑏 𝑥 <s 𝑦 ↔ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦))
96, 83anbi23d 1436 . . 3 (𝑏 = 𝐵 → ((𝐴 No 𝑏 No ∧ ∀𝑥𝐴𝑦𝑏 𝑥 <s 𝑦) ↔ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)))
105, 9, 1brabg 5391 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 <<s 𝐵 ↔ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)))
112, 10biadanii 821 1 (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ⊆ wss 3881   class class class wbr 5030   No csur 33260
 Copyright terms: Public domain W3C validator