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Theorem ssltun2 27869
Description: Union law for surreal set less-than. (Contributed by Scott Fenton, 9-Dec-2021.)
Assertion
Ref Expression
ssltun2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 <<s (𝐵𝐶))

Proof of Theorem ssltun2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssltex1 27846 . . 3 (𝐴 <<s 𝐵𝐴 ∈ V)
21adantr 480 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 ∈ V)
3 ssltex2 27847 . . . 4 (𝐴 <<s 𝐵𝐵 ∈ V)
43adantr 480 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐵 ∈ V)
5 ssltex2 27847 . . . 4 (𝐴 <<s 𝐶𝐶 ∈ V)
65adantl 481 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐶 ∈ V)
74, 6unexd 7773 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐵𝐶) ∈ V)
8 ssltss1 27848 . . 3 (𝐴 <<s 𝐵𝐴 No )
98adantr 480 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 No )
10 ssltss2 27849 . . . 4 (𝐴 <<s 𝐵𝐵 No )
1110adantr 480 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐵 No )
12 ssltss2 27849 . . . 4 (𝐴 <<s 𝐶𝐶 No )
1312adantl 481 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐶 No )
1411, 13unssd 4202 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐵𝐶) ⊆ No )
15 elun 4163 . . . 4 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
16 ssltsepc 27853 . . . . . . . 8 ((𝐴 <<s 𝐵𝑥𝐴𝑦𝐵) → 𝑥 <s 𝑦)
17163exp 1118 . . . . . . 7 (𝐴 <<s 𝐵 → (𝑥𝐴 → (𝑦𝐵𝑥 <s 𝑦)))
1817adantr 480 . . . . . 6 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝑥𝐴 → (𝑦𝐵𝑥 <s 𝑦)))
1918com3r 87 . . . . 5 (𝑦𝐵 → ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝑥𝐴𝑥 <s 𝑦)))
20 ssltsepc 27853 . . . . . . . 8 ((𝐴 <<s 𝐶𝑥𝐴𝑦𝐶) → 𝑥 <s 𝑦)
21203exp 1118 . . . . . . 7 (𝐴 <<s 𝐶 → (𝑥𝐴 → (𝑦𝐶𝑥 <s 𝑦)))
2221adantl 481 . . . . . 6 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝑥𝐴 → (𝑦𝐶𝑥 <s 𝑦)))
2322com3r 87 . . . . 5 (𝑦𝐶 → ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝑥𝐴𝑥 <s 𝑦)))
2419, 23jaoi 857 . . . 4 ((𝑦𝐵𝑦𝐶) → ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝑥𝐴𝑥 <s 𝑦)))
2515, 24sylbi 217 . . 3 (𝑦 ∈ (𝐵𝐶) → ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝑥𝐴𝑥 <s 𝑦)))
26253imp231 1112 . 2 (((𝐴 <<s 𝐵𝐴 <<s 𝐶) ∧ 𝑥𝐴𝑦 ∈ (𝐵𝐶)) → 𝑥 <s 𝑦)
272, 7, 9, 14, 26ssltd 27851 1 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 <<s (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  wcel 2106  Vcvv 3478  cun 3961  wss 3963   class class class wbr 5148   No csur 27699   <s cslt 27700   <<s csslt 27840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-sslt 27841
This theorem is referenced by:  scutun12  27870
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