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Theorem sssslt2 27679
Description: Relation between surreal set less-than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
Assertion
Ref Expression
sssslt2 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 <<s 𝐶)

Proof of Theorem sssslt2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssltex1 27669 . . 3 (𝐴 <<s 𝐵𝐴 ∈ V)
21adantr 480 . 2 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 ∈ V)
3 ssltex2 27670 . . . 4 (𝐴 <<s 𝐵𝐵 ∈ V)
43adantr 480 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐵 ∈ V)
5 simpr 484 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐶𝐵)
64, 5ssexd 5317 . 2 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐶 ∈ V)
7 ssltss1 27671 . . . 4 (𝐴 <<s 𝐵𝐴 No )
87adantr 480 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 No )
9 ssltss2 27672 . . . . 5 (𝐴 <<s 𝐵𝐵 No )
109adantr 480 . . . 4 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐵 No )
115, 10sstrd 3987 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐶 No )
12 ssltsep 27673 . . . 4 (𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
13 ssralv 4045 . . . . 5 (𝐶𝐵 → (∀𝑦𝐵 𝑥 <s 𝑦 → ∀𝑦𝐶 𝑥 <s 𝑦))
1413ralimdv 3163 . . . 4 (𝐶𝐵 → (∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦 → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦))
1512, 14mpan9 506 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)
168, 11, 153jca 1125 . 2 ((𝐴 <<s 𝐵𝐶𝐵) → (𝐴 No 𝐶 No ∧ ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦))
17 brsslt 27668 . 2 (𝐴 <<s 𝐶 ↔ ((𝐴 ∈ V ∧ 𝐶 ∈ V) ∧ (𝐴 No 𝐶 No ∧ ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)))
182, 6, 16, 17syl21anbrc 1341 1 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 <<s 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084  wcel 2098  wral 3055  Vcvv 3468  wss 3943   class class class wbr 5141   No csur 27523   <s cslt 27524   <<s csslt 27663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-sslt 27664
This theorem is referenced by:  scutun12  27693
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