Theorem brslts 28130

Description: Binary relation form of the surreal set less-than relation. (Contributed by Scott Fenton, 8-Dec-2021.)

Assertion

Ref	Expression
brslts	⊢ (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 brslts

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-slts 28126	. . 3 ⊢ <<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦)}
2	1	bropaex12 5742	. 2 ⊢ (𝐴 <<s 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		sseq1 3956	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ No ↔ 𝐴 ⊆ No ))
4		raleq 3317	. . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦))
5	3, 4	3anbi13d 1466	. . 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 3956	. . . 4 ⊢ (𝑏 = 𝐵 → (𝑏 ⊆ No ↔ 𝐵 ⊆ No ))
7		raleq 3317	. . . . 5 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 𝑥 <s 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))
8	7	ralbidv 3186	. . . 4 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))
9	6, 8	3anbi23d 1467	. . 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 5514	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 <<s 𝐵 ↔ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)))
11	2, 10	biadanii 834	1 ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451   ⊆ wss 3899   class class class wbr 5103   No csur 27979   <s clts 27980   <<s cslts 28125

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-slts 28126

This theorem is used by:  sltsex1  28131  sltsex2  28132  sltsss1  28133  sltsss2  28134  sltssep  28135  sltsd  28136  sltssnb  28137  ssslts1  28141  ssslts2  28142  conway  28147  etaslts  28161  lesrec  28167  cofcutr  28292