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Theorem sssslt1 27118
Description: Relation 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 27110 . . . 4 (𝐴 <<s 𝐵𝐴 ∈ V)
21adantr 481 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐴 ∈ V)
3 simpr 485 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶𝐴)
42, 3ssexd 5279 . 2 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 ∈ V)
5 ssltex2 27111 . . 3 (𝐴 <<s 𝐵𝐵 ∈ V)
65adantr 481 . 2 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐵 ∈ V)
7 ssltss1 27112 . . . . 5 (𝐴 <<s 𝐵𝐴 No )
87adantr 481 . . . 4 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐴 No )
93, 8sstrd 3952 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 No )
10 ssltss2 27113 . . . 4 (𝐴 <<s 𝐵𝐵 No )
1110adantr 481 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐵 No )
12 ssltsep 27114 . . . 4 (𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
13 ssralv 4008 . . . 4 (𝐶𝐴 → (∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦 → ∀𝑥𝐶𝑦𝐵 𝑥 <s 𝑦))
1412, 13mpan9 507 . . 3 ((𝐴 <<s 𝐵𝐶𝐴) → ∀𝑥𝐶𝑦𝐵 𝑥 <s 𝑦)
159, 11, 143jca 1128 . 2 ((𝐴 <<s 𝐵𝐶𝐴) → (𝐶 No 𝐵 No ∧ ∀𝑥𝐶𝑦𝐵 𝑥 <s 𝑦))
16 brsslt 27109 . 2 (𝐶 <<s 𝐵 ↔ ((𝐶 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 No 𝐵 No ∧ ∀𝑥𝐶𝑦𝐵 𝑥 <s 𝑦)))
174, 6, 15, 16syl21anbrc 1344 1 ((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 <<s 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087  wcel 2106  wral 3062  Vcvv 3443  wss 3908   class class class wbr 5103   No csur 26972   <s cslt 26973   <<s csslt 27104
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-xp 5637  df-sslt 27105
This theorem is referenced by:  scutun12  27133
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