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Theorem brsslt 33151
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 33148 . . 3 <<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 No 𝑏 No ∧ ∀𝑥𝑎𝑦𝑏 𝑥 <s 𝑦)}
21bropaex12 5635 . 2 (𝐴 <<s 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 sseq1 3989 . . . 4 (𝑎 = 𝐴 → (𝑎 No 𝐴 No ))
4 raleq 3403 . . . 4 (𝑎 = 𝐴 → (∀𝑥𝑎𝑦𝑏 𝑥 <s 𝑦 ↔ ∀𝑥𝐴𝑦𝑏 𝑥 <s 𝑦))
53, 43anbi13d 1429 . . 3 (𝑎 = 𝐴 → ((𝑎 No 𝑏 No ∧ ∀𝑥𝑎𝑦𝑏 𝑥 <s 𝑦) ↔ (𝐴 No 𝑏 No ∧ ∀𝑥𝐴𝑦𝑏 𝑥 <s 𝑦)))
6 sseq1 3989 . . . 4 (𝑏 = 𝐵 → (𝑏 No 𝐵 No ))
7 raleq 3403 . . . . 5 (𝑏 = 𝐵 → (∀𝑦𝑏 𝑥 <s 𝑦 ↔ ∀𝑦𝐵 𝑥 <s 𝑦))
87ralbidv 3194 . . . 4 (𝑏 = 𝐵 → (∀𝑥𝐴𝑦𝑏 𝑥 <s 𝑦 ↔ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦))
96, 83anbi23d 1430 . . 3 (𝑏 = 𝐵 → ((𝐴 No 𝑏 No ∧ ∀𝑥𝐴𝑦𝑏 𝑥 <s 𝑦) ↔ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)))
105, 9, 1brabg 5417 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 <<s 𝐵 ↔ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)))
112, 10biadanii 818 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  wss 3933   class class class wbr 5057   No csur 33044   <s cslt 33045   <<s csslt 33147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-sslt 33148
This theorem is referenced by:  ssltex1  33152  ssltex2  33153  ssltss1  33154  ssltss2  33155  ssltsep  33156  sssslt1  33157  sssslt2  33158  nulsslt  33159  nulssgt  33160  conway  33161  sslttr  33165  ssltun1  33166  ssltun2  33167  etasslt  33171  slerec  33174
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