Theorem ssslts1 28141

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 28131	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
2	1	adantr 486	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐴 ∈ V)
3		simpr 490	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ⊆ 𝐴)
4	2, 3	ssexd 5286	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ∈ V)
5		sltsex2 28132	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
6	5	adantr 486	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐵 ∈ V)
7		sltsss1 28133	. . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
8	7	adantr 486	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐴 ⊆ No )
9	3, 8	sstrd 3941	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ⊆ No )
10		sltsss2 28134	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
11	10	adantr 486	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐵 ⊆ No )
12		sltssep 28135	. . . 4 ⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)
13		ssralv 4000	. . . 4 ⊢ (𝐶 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))
14	12, 13	mpan9 516	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)
15	9, 11, 14	3jca 1146	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐶 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))
16		brslts 28130	. 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 1363	1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 <<s 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ 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:  cutsun12  28158  eqcuts3  28172  cutmax  28302  precsexlem11  28585