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Theorem ssltun2 27762
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 27739 . . 3 (𝐴 <<s 𝐵𝐴 ∈ V)
21adantr 479 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 ∈ V)
3 ssltex2 27740 . . . 4 (𝐴 <<s 𝐵𝐵 ∈ V)
43adantr 479 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐵 ∈ V)
5 ssltex2 27740 . . . 4 (𝐴 <<s 𝐶𝐶 ∈ V)
65adantl 480 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐶 ∈ V)
74, 6unexd 7762 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐵𝐶) ∈ V)
8 ssltss1 27741 . . 3 (𝐴 <<s 𝐵𝐴 No )
98adantr 479 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 No )
10 ssltss2 27742 . . . 4 (𝐴 <<s 𝐵𝐵 No )
1110adantr 479 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐵 No )
12 ssltss2 27742 . . . 4 (𝐴 <<s 𝐶𝐶 No )
1312adantl 480 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐶 No )
1411, 13unssd 4188 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐵𝐶) ⊆ No )
15 elun 4149 . . . 4 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
16 ssltsepc 27746 . . . . . . . 8 ((𝐴 <<s 𝐵𝑥𝐴𝑦𝐵) → 𝑥 <s 𝑦)
17163exp 1116 . . . . . . 7 (𝐴 <<s 𝐵 → (𝑥𝐴 → (𝑦𝐵𝑥 <s 𝑦)))
1817adantr 479 . . . . . 6 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝑥𝐴 → (𝑦𝐵𝑥 <s 𝑦)))
1918com3r 87 . . . . 5 (𝑦𝐵 → ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝑥𝐴𝑥 <s 𝑦)))
20 ssltsepc 27746 . . . . . . . 8 ((𝐴 <<s 𝐶𝑥𝐴𝑦𝐶) → 𝑥 <s 𝑦)
21203exp 1116 . . . . . . 7 (𝐴 <<s 𝐶 → (𝑥𝐴 → (𝑦𝐶𝑥 <s 𝑦)))
2221adantl 480 . . . . . 6 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝑥𝐴 → (𝑦𝐶𝑥 <s 𝑦)))
2322com3r 87 . . . . 5 (𝑦𝐶 → ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝑥𝐴𝑥 <s 𝑦)))
2419, 23jaoi 855 . . . 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 1110 . 2 (((𝐴 <<s 𝐵𝐴 <<s 𝐶) ∧ 𝑥𝐴𝑦 ∈ (𝐵𝐶)) → 𝑥 <s 𝑦)
272, 7, 9, 14, 26ssltd 27744 1 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 <<s (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wo 845  wcel 2098  Vcvv 3473  cun 3947  wss 3949   class class class wbr 5152   No csur 27593   <s cslt 27594   <<s csslt 27733
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-xp 5688  df-sslt 27734
This theorem is referenced by:  scutun12  27763
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