Theorem ssslts1 27765

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

Assertion

Ref	Expression
ssslts1	⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 <<s 𝐵)

Proof of Theorem ssslts1

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

Step	Hyp	Ref	Expression
1		sltsex1 27755	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
2	1	adantr 480	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐴 ∈ V)
3		simpr 484	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ⊆ 𝐴)
4	2, 3	ssexd 5265	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ∈ V)
5		sltsex2 27756	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
6	5	adantr 480	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐵 ∈ V)
7		sltsss1 27757	. . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
8	7	adantr 480	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐴 ⊆ No )
9	3, 8	sstrd 3932	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ⊆ No )
10		sltsss2 27758	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
11	10	adantr 480	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐵 ⊆ No )
12		sltssep 27759	. . . 4 ⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)
13		ssralv 3990	. . . 4 ⊢ (𝐶 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))
14	12, 13	mpan9 506	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)
15	9, 11, 14	3jca 1129	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐶 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))
16		brslts 27754	. 2 ⊢ (𝐶 <<s 𝐵 ↔ ((𝐶 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)))
17	4, 6, 15, 16	syl21anbrc 1346	1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 <<s 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2114  ∀wral 3051  Vcvv 3429   ⊆ wss 3889   class class class wbr 5085   No csur 27603   <s clts 27604   <<s cslts 27749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-slts 27750

This theorem is referenced by:  cutsun12  27782  eqcuts3  27796  cutmax  27926  precsexlem11  28209