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Theorem ssltun2 32797
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 32782 . . . 4 (𝐴 <<s 𝐵𝐴 ∈ V)
21adantr 473 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 ∈ V)
3 ssltex2 32783 . . . 4 (𝐴 <<s 𝐵𝐵 ∈ V)
4 ssltex2 32783 . . . 4 (𝐴 <<s 𝐶𝐶 ∈ V)
5 unexg 7289 . . . 4 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐵𝐶) ∈ V)
63, 4, 5syl2an 586 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐵𝐶) ∈ V)
72, 6jca 504 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐴 ∈ V ∧ (𝐵𝐶) ∈ V))
8 ssltss1 32784 . . . 4 (𝐴 <<s 𝐵𝐴 No )
98adantr 473 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 No )
10 ssltss2 32785 . . . . 5 (𝐴 <<s 𝐵𝐵 No )
1110adantr 473 . . . 4 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐵 No )
12 ssltss2 32785 . . . . 5 (𝐴 <<s 𝐶𝐶 No )
1312adantl 474 . . . 4 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐶 No )
1411, 13unssd 4050 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐵𝐶) ⊆ No )
15 ssltsep 32786 . . . . 5 (𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
1615adantr 473 . . . 4 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
17 ssltsep 32786 . . . . 5 (𝐴 <<s 𝐶 → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)
1817adantl 474 . . . 4 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)
19 ralunb 4055 . . . . . 6 (∀𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦 ↔ (∀𝑦𝐵 𝑥 <s 𝑦 ∧ ∀𝑦𝐶 𝑥 <s 𝑦))
2019ralbii 3115 . . . . 5 (∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦 ↔ ∀𝑥𝐴 (∀𝑦𝐵 𝑥 <s 𝑦 ∧ ∀𝑦𝐶 𝑥 <s 𝑦))
21 r19.26 3120 . . . . 5 (∀𝑥𝐴 (∀𝑦𝐵 𝑥 <s 𝑦 ∧ ∀𝑦𝐶 𝑥 <s 𝑦) ↔ (∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦 ∧ ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦))
2220, 21bitri 267 . . . 4 (∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦 ↔ (∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦 ∧ ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦))
2316, 18, 22sylanbrc 575 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → ∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦)
249, 14, 233jca 1108 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐴 No ∧ (𝐵𝐶) ⊆ No ∧ ∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦))
25 brsslt 32781 . 2 (𝐴 <<s (𝐵𝐶) ↔ ((𝐴 ∈ V ∧ (𝐵𝐶) ∈ V) ∧ (𝐴 No ∧ (𝐵𝐶) ⊆ No ∧ ∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦)))
267, 24, 25sylanbrc 575 1 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 <<s (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068  wcel 2050  wral 3088  Vcvv 3415  cun 3827  wss 3829   class class class wbr 4929   No csur 32674   <s cslt 32675   <<s csslt 32777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-xp 5413  df-sslt 32778
This theorem is referenced by:  scutun12  32798
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