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Theorem sssslt2 33371
Description: Relationship 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 33365 . . 3 (𝐴 <<s 𝐵𝐴 ∈ V)
21adantr 484 . 2 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 ∈ V)
3 ssltex2 33366 . . . 4 (𝐴 <<s 𝐵𝐵 ∈ V)
43adantr 484 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐵 ∈ V)
5 simpr 488 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐶𝐵)
64, 5ssexd 5192 . 2 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐶 ∈ V)
7 ssltss1 33367 . . . 4 (𝐴 <<s 𝐵𝐴 No )
87adantr 484 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 No )
9 ssltss2 33368 . . . . 5 (𝐴 <<s 𝐵𝐵 No )
109adantr 484 . . . 4 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐵 No )
115, 10sstrd 3925 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐶 No )
12 ssltsep 33369 . . . . 5 (𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
1312adantr 484 . . . 4 ((𝐴 <<s 𝐵𝐶𝐵) → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
14 ssralv 3981 . . . . . 6 (𝐶𝐵 → (∀𝑦𝐵 𝑥 <s 𝑦 → ∀𝑦𝐶 𝑥 <s 𝑦))
155, 14syl 17 . . . . 5 ((𝐴 <<s 𝐵𝐶𝐵) → (∀𝑦𝐵 𝑥 <s 𝑦 → ∀𝑦𝐶 𝑥 <s 𝑦))
1615ralimdv 3145 . . . 4 ((𝐴 <<s 𝐵𝐶𝐵) → (∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦 → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦))
1713, 16mpd 15 . . 3 ((𝐴 <<s 𝐵𝐶𝐵) → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)
188, 11, 173jca 1125 . 2 ((𝐴 <<s 𝐵𝐶𝐵) → (𝐴 No 𝐶 No ∧ ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦))
19 brsslt 33364 . 2 (𝐴 <<s 𝐶 ↔ ((𝐴 ∈ V ∧ 𝐶 ∈ V) ∧ (𝐴 No 𝐶 No ∧ ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)))
202, 6, 18, 19syl21anbrc 1341 1 ((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 <<s 𝐶)
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 33257   <s cslt 33258   <<s csslt 33360
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-sslt 33361
This theorem is referenced by:  scutun12  33381
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