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Theorem ssltun2 27170
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 27148 . . 3 (𝐴 <<s 𝐵𝐴 ∈ V)
21adantr 482 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 ∈ V)
3 ssltex2 27149 . . . 4 (𝐴 <<s 𝐵𝐵 ∈ V)
43adantr 482 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐵 ∈ V)
5 ssltex2 27149 . . . 4 (𝐴 <<s 𝐶𝐶 ∈ V)
65adantl 483 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐶 ∈ V)
74, 6unexd 7689 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐵𝐶) ∈ V)
8 ssltss1 27150 . . 3 (𝐴 <<s 𝐵𝐴 No )
98adantr 482 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 No )
10 ssltss2 27151 . . . 4 (𝐴 <<s 𝐵𝐵 No )
1110adantr 482 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐵 No )
12 ssltss2 27151 . . . 4 (𝐴 <<s 𝐶𝐶 No )
1312adantl 483 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐶 No )
1411, 13unssd 4147 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐵𝐶) ⊆ No )
15 elun 4109 . . . 4 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
16 ssltsepc 27154 . . . . . . . 8 ((𝐴 <<s 𝐵𝑥𝐴𝑦𝐵) → 𝑥 <s 𝑦)
17163exp 1120 . . . . . . 7 (𝐴 <<s 𝐵 → (𝑥𝐴 → (𝑦𝐵𝑥 <s 𝑦)))
1817adantr 482 . . . . . 6 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝑥𝐴 → (𝑦𝐵𝑥 <s 𝑦)))
1918com3r 87 . . . . 5 (𝑦𝐵 → ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝑥𝐴𝑥 <s 𝑦)))
20 ssltsepc 27154 . . . . . . . 8 ((𝐴 <<s 𝐶𝑥𝐴𝑦𝐶) → 𝑥 <s 𝑦)
21203exp 1120 . . . . . . 7 (𝐴 <<s 𝐶 → (𝑥𝐴 → (𝑦𝐶𝑥 <s 𝑦)))
2221adantl 483 . . . . . 6 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝑥𝐴 → (𝑦𝐶𝑥 <s 𝑦)))
2322com3r 87 . . . . 5 (𝑦𝐶 → ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝑥𝐴𝑥 <s 𝑦)))
2419, 23jaoi 856 . . . 4 ((𝑦𝐵𝑦𝐶) → ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝑥𝐴𝑥 <s 𝑦)))
2515, 24sylbi 216 . . 3 (𝑦 ∈ (𝐵𝐶) → ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝑥𝐴𝑥 <s 𝑦)))
26253imp231 1114 . 2 (((𝐴 <<s 𝐵𝐴 <<s 𝐶) ∧ 𝑥𝐴𝑦 ∈ (𝐵𝐶)) → 𝑥 <s 𝑦)
272, 7, 9, 14, 26ssltd 27153 1 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 <<s (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wo 846  wcel 2107  Vcvv 3444  cun 3909  wss 3911   class class class wbr 5106   No csur 27004   <s cslt 27005   <<s csslt 27142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-sslt 27143
This theorem is referenced by:  scutun12  27171
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