Theorem brsslt 27812

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 27808	. . 3 ⊢ <<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦)}
2	1	bropaex12 5775	. 2 ⊢ (𝐴 <<s 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		sseq1 4016	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ No ↔ 𝐴 ⊆ No ))
4		raleq 3316	. . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦))
5	3, 4	3anbi13d 1435	. . 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 4016	. . . 4 ⊢ (𝑏 = 𝐵 → (𝑏 ⊆ No ↔ 𝐵 ⊆ No ))
7		raleq 3316	. . . . 5 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 𝑥 <s 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))
8	7	ralbidv 3171	. . . 4 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))
9	6, 8	3anbi23d 1436	. . 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 5547	. 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 205   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2100  ∀wral 3054  Vcvv 3472   ⊆ wss 3958   class class class wbr 5154   No csur 27666   <s cslt 27667   <<s csslt 27807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-ext 2700  ax-sep 5305  ax-nul 5312  ax-pr 5435

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2062  df-clab 2707  df-cleq 2721  df-clel 2806  df-ral 3055  df-rex 3064  df-rab 3429  df-v 3474  df-dif 3961  df-un 3963  df-ss 3975  df-nul 4334  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-br 5155  df-opab 5217  df-xp 5690  df-sslt 27808

This theorem is referenced by:  ssltex1  27813  ssltex2  27814  ssltss1  27815  ssltss2  27816  ssltsep  27817  ssltd  27818  sssslt1  27822  sssslt2  27823  conway  27826  etasslt  27840  slerec  27846  cofcutr  27938