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Theorem brsslt 27848
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

Proof of Theorem brsslt
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sslt 27844 . . 3 <<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 No 𝑏 No ∧ ∀𝑥𝑎𝑦𝑏 𝑥 <s 𝑦)}
21bropaex12 5791 . 2 (𝐴 <<s 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 sseq1 4034 . . . 4 (𝑎 = 𝐴 → (𝑎 No 𝐴 No ))
4 raleq 3331 . . . 4 (𝑎 = 𝐴 → (∀𝑥𝑎𝑦𝑏 𝑥 <s 𝑦 ↔ ∀𝑥𝐴𝑦𝑏 𝑥 <s 𝑦))
53, 43anbi13d 1438 . . 3 (𝑎 = 𝐴 → ((𝑎 No 𝑏 No ∧ ∀𝑥𝑎𝑦𝑏 𝑥 <s 𝑦) ↔ (𝐴 No 𝑏 No ∧ ∀𝑥𝐴𝑦𝑏 𝑥 <s 𝑦)))
6 sseq1 4034 . . . 4 (𝑏 = 𝐵 → (𝑏 No 𝐵 No ))
7 raleq 3331 . . . . 5 (𝑏 = 𝐵 → (∀𝑦𝑏 𝑥 <s 𝑦 ↔ ∀𝑦𝐵 𝑥 <s 𝑦))
87ralbidv 3184 . . . 4 (𝑏 = 𝐵 → (∀𝑥𝐴𝑦𝑏 𝑥 <s 𝑦 ↔ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦))
96, 83anbi23d 1439 . . 3 (𝑏 = 𝐵 → ((𝐴 No 𝑏 No ∧ ∀𝑥𝐴𝑦𝑏 𝑥 <s 𝑦) ↔ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)))
105, 9, 1brabg 5558 . 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 206  wa 395  w3a 1087   = wceq 1537  wcel 2108  wral 3067  Vcvv 3488  wss 3976   class class class wbr 5166   No csur 27702   <s cslt 27703   <<s csslt 27843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-sslt 27844
This theorem is referenced by:  ssltex1  27849  ssltex2  27850  ssltss1  27851  ssltss2  27852  ssltsep  27853  ssltd  27854  sssslt1  27858  sssslt2  27859  conway  27862  etasslt  27876  slerec  27882  cofcutr  27976
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