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Theorem sssslt1 33385
 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 33380 . . . 4 (𝐴 <<s 𝐵𝐴 ∈ V)
21adantr 484 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐴 ∈ V)
3 simpr 488 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶𝐴)
42, 3ssexd 5192 . 2 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 ∈ V)
5 ssltex2 33381 . . 3 (𝐴 <<s 𝐵𝐵 ∈ V)
65adantr 484 . 2 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐵 ∈ V)
7 ssltss1 33382 . . . . 5 (𝐴 <<s 𝐵𝐴 No )
87adantr 484 . . . 4 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐴 No )
93, 8sstrd 3925 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 No )
10 ssltss2 33383 . . . 4 (𝐴 <<s 𝐵𝐵 No )
1110adantr 484 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐵 No )
12 ssltsep 33384 . . . 4 (𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
13 ssralv 3981 . . . 4 (𝐶𝐴 → (∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦 → ∀𝑥𝐶𝑦𝐵 𝑥 <s 𝑦))
1412, 13mpan9 510 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → ∀𝑥𝐶𝑦𝐵 𝑥 <s 𝑦)
159, 11, 143jca 1125 . 2 ((𝐴 <<s 𝐵𝐶𝐴) → (𝐶 No 𝐵 No ∧ ∀𝑥𝐶𝑦𝐵 𝑥 <s 𝑦))
16 brsslt 33379 . 2 (𝐶 <<s 𝐵 ↔ ((𝐶 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 No 𝐵 No ∧ ∀𝑥𝐶𝑦𝐵 𝑥 <s 𝑦)))
174, 6, 15, 16syl21anbrc 1341 1 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 <<s 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ⊆ wss 3881   class class class wbr 5030   No csur 33272
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