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Theorem ssltun2 33999
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 33977 . . 3 (𝐴 <<s 𝐵𝐴 ∈ V)
21adantr 481 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 ∈ V)
3 ssltex2 33978 . . . 4 (𝐴 <<s 𝐵𝐵 ∈ V)
43adantr 481 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐵 ∈ V)
5 ssltex2 33978 . . . 4 (𝐴 <<s 𝐶𝐶 ∈ V)
65adantl 482 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐶 ∈ V)
74, 6unexd 7598 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐵𝐶) ∈ V)
8 ssltss1 33979 . . 3 (𝐴 <<s 𝐵𝐴 No )
98adantr 481 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 No )
10 ssltss2 33980 . . . 4 (𝐴 <<s 𝐵𝐵 No )
1110adantr 481 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐵 No )
12 ssltss2 33980 . . . 4 (𝐴 <<s 𝐶𝐶 No )
1312adantl 482 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐶 No )
1411, 13unssd 4125 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐵𝐶) ⊆ No )
15 elun 4088 . . . 4 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
16 ssltsepc 33983 . . . . . . . 8 ((𝐴 <<s 𝐵𝑥𝐴𝑦𝐵) → 𝑥 <s 𝑦)
17163exp 1118 . . . . . . 7 (𝐴 <<s 𝐵 → (𝑥𝐴 → (𝑦𝐵𝑥 <s 𝑦)))
1817adantr 481 . . . . . 6 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝑥𝐴 → (𝑦𝐵𝑥 <s 𝑦)))
1918com3r 87 . . . . 5 (𝑦𝐵 → ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝑥𝐴𝑥 <s 𝑦)))
20 ssltsepc 33983 . . . . . . . 8 ((𝐴 <<s 𝐶𝑥𝐴𝑦𝐶) → 𝑥 <s 𝑦)
21203exp 1118 . . . . . . 7 (𝐴 <<s 𝐶 → (𝑥𝐴 → (𝑦𝐶𝑥 <s 𝑦)))
2221adantl 482 . . . . . 6 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝑥𝐴 → (𝑦𝐶𝑥 <s 𝑦)))
2322com3r 87 . . . . 5 (𝑦𝐶 → ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝑥𝐴𝑥 <s 𝑦)))
2419, 23jaoi 854 . . . 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 1112 . 2 (((𝐴 <<s 𝐵𝐴 <<s 𝐶) ∧ 𝑥𝐴𝑦 ∈ (𝐵𝐶)) → 𝑥 <s 𝑦)
272, 7, 9, 14, 26ssltd 33982 1 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 <<s (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 844  wcel 2110  Vcvv 3431  cun 3890  wss 3892   class class class wbr 5079   No csur 33839   <s cslt 33840   <<s csslt 33971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-xp 5596  df-sslt 33972
This theorem is referenced by:  scutun12  34000
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