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Theorem brsslt 27077
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 27073 . . 3 <<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 No 𝑏 No ∧ ∀𝑥𝑎𝑦𝑏 𝑥 <s 𝑦)}
21bropaex12 5722 . 2 (𝐴 <<s 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 sseq1 3968 . . . 4 (𝑎 = 𝐴 → (𝑎 No 𝐴 No ))
4 raleq 3308 . . . 4 (𝑎 = 𝐴 → (∀𝑥𝑎𝑦𝑏 𝑥 <s 𝑦 ↔ ∀𝑥𝐴𝑦𝑏 𝑥 <s 𝑦))
53, 43anbi13d 1439 . . 3 (𝑎 = 𝐴 → ((𝑎 No 𝑏 No ∧ ∀𝑥𝑎𝑦𝑏 𝑥 <s 𝑦) ↔ (𝐴 No 𝑏 No ∧ ∀𝑥𝐴𝑦𝑏 𝑥 <s 𝑦)))
6 sseq1 3968 . . . 4 (𝑏 = 𝐵 → (𝑏 No 𝐵 No ))
7 raleq 3308 . . . . 5 (𝑏 = 𝐵 → (∀𝑦𝑏 𝑥 <s 𝑦 ↔ ∀𝑦𝐵 𝑥 <s 𝑦))
87ralbidv 3173 . . . 4 (𝑏 = 𝐵 → (∀𝑥𝐴𝑦𝑏 𝑥 <s 𝑦 ↔ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦))
96, 83anbi23d 1440 . . 3 (𝑏 = 𝐵 → ((𝐴 No 𝑏 No ∧ ∀𝑥𝐴𝑦𝑏 𝑥 <s 𝑦) ↔ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)))
105, 9, 1brabg 5495 . 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3063  Vcvv 3444  wss 3909   class class class wbr 5104   No csur 26940   <s cslt 26941   <<s csslt 27072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-xp 5638  df-sslt 27073
This theorem is referenced by:  ssltex1  27078  ssltex2  27079  ssltss1  27080  ssltss2  27081  ssltsep  27082  ssltd  27083  sssslt1  27086  sssslt2  27087  conway  27090  etasslt  27104  slerec  27110  cofcutr  27192
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