Theorem sssslt2 33990

Description: Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)

Assertion

Ref	Expression
sssslt2	⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 <<s 𝐶)

Proof of Theorem sssslt2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssltex1 33981	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
2	1	adantr 481	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 ∈ V)
3		ssltex2 33982	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
4	3	adantr 481	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐵 ∈ V)
5		simpr 485	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ 𝐵)
6	4, 5	ssexd 5248	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ∈ V)
7		ssltss1 33983	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
8	7	adantr 481	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 ⊆ No )
9		ssltss2 33984	. . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
10	9	adantr 481	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐵 ⊆ No )
11	5, 10	sstrd 3931	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ No )
12		ssltsep 33985	. . . 4 ⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)
13		ssralv 3987	. . . . 5 ⊢ (𝐶 ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 → ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦))
14	13	ralimdv 3109	. . . 4 ⊢ (𝐶 ⊆ 𝐵 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦))
15	12, 14	mpan9 507	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)
16	8, 11, 15	3jca 1127	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐴 ⊆ No ∧ 𝐶 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦))
17		brsslt 33980	. 2 ⊢ (𝐴 <<s 𝐶 ↔ ((𝐴 ∈ V ∧ 𝐶 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐶 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)))
18	2, 6, 16, 17	syl21anbrc 1343	1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 <<s 𝐶)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ⊆ wss 3887   class class class wbr 5074   No csur 33843   <s cslt 33844   <<s csslt 33975

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-sslt 33976

This theorem is referenced by:  scutun12  34004