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Theorem sssslt1 32243
 Description: Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
Assertion
Ref Expression
sssslt1 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 <<s 𝐵)

Proof of Theorem sssslt1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssltex1 32238 . . . . 5 (𝐴 <<s 𝐵𝐴 ∈ V)
21adantr 466 . . . 4 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐴 ∈ V)
3 simpr 471 . . . 4 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶𝐴)
42, 3ssexd 4939 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 ∈ V)
5 ssltex2 32239 . . . 4 (𝐴 <<s 𝐵𝐵 ∈ V)
65adantr 466 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐵 ∈ V)
74, 6jca 501 . 2 ((𝐴 <<s 𝐵𝐶𝐴) → (𝐶 ∈ V ∧ 𝐵 ∈ V))
8 ssltss1 32240 . . . . 5 (𝐴 <<s 𝐵𝐴 No )
98adantr 466 . . . 4 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐴 No )
103, 9sstrd 3762 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 No )
11 ssltss2 32241 . . . 4 (𝐴 <<s 𝐵𝐵 No )
1211adantr 466 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐵 No )
13 ssltsep 32242 . . . 4 (𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
14 ssralv 3815 . . . 4 (𝐶𝐴 → (∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦 → ∀𝑥𝐶𝑦𝐵 𝑥 <s 𝑦))
1513, 14mpan9 496 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → ∀𝑥𝐶𝑦𝐵 𝑥 <s 𝑦)
1610, 12, 153jca 1122 . 2 ((𝐴 <<s 𝐵𝐶𝐴) → (𝐶 No 𝐵 No ∧ ∀𝑥𝐶𝑦𝐵 𝑥 <s 𝑦))
17 brsslt 32237 . 2 (𝐶 <<s 𝐵 ↔ ((𝐶 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 No 𝐵 No ∧ ∀𝑥𝐶𝑦𝐵 𝑥 <s 𝑦)))
187, 16, 17sylanbrc 572 1 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 <<s 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1071   ∈ wcel 2145  ∀wral 3061  Vcvv 3351   ⊆ wss 3723   class class class wbr 4786   No csur 32130
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