Theorem brsslt 33254

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 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 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.

Step	Hyp	Ref	Expression
1		df-sslt 33251	. . 3 ⊢ <<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦)}
2	1	bropaex12 5641	. 2 ⊢ (𝐴 <<s 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		sseq1 3991	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ No ↔ 𝐴 ⊆ No ))
4		raleq 3405	. . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦))
5	3, 4	3anbi13d 1434	. . 3 ⊢ (𝑎 = 𝐴 → ((𝑎 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦) ↔ (𝐴 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦)))
6		sseq1 3991	. . . 4 ⊢ (𝑏 = 𝐵 → (𝑏 ⊆ No ↔ 𝐵 ⊆ No ))
7		raleq 3405	. . . . 5 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 𝑥 <s 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))
8	7	ralbidv 3197	. . . 4 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))
9	6, 8	3anbi23d 1435	. . 3 ⊢ (𝑏 = 𝐵 → ((𝐴 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦) ↔ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)))
10	5, 9, 1	brabg 5425	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 <<s 𝐵 ↔ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)))
11	2, 10	biadanii 820	1 ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3494   ⊆ wss 3935   class class class wbr 5065   No csur 33147   <s cslt 33148   <<s csslt 33250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-xp 5560  df-sslt 33251

This theorem is referenced by:  ssltex1  33255  ssltex2  33256  ssltss1  33257  ssltss2  33258  ssltsep  33259  sssslt1  33260  sssslt2  33261  nulsslt  33262  nulssgt  33263  conway  33264  sslttr  33268  ssltun1  33269  ssltun2  33270  etasslt  33274  slerec  33277