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Theorem ssltun2 33344
 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 33329 . . . 4 (𝐴 <<s 𝐵𝐴 ∈ V)
21adantr 484 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 ∈ V)
3 ssltex2 33330 . . . 4 (𝐴 <<s 𝐵𝐵 ∈ V)
4 ssltex2 33330 . . . 4 (𝐴 <<s 𝐶𝐶 ∈ V)
5 unexg 7457 . . . 4 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐵𝐶) ∈ V)
63, 4, 5syl2an 598 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐵𝐶) ∈ V)
72, 6jca 515 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐴 ∈ V ∧ (𝐵𝐶) ∈ V))
8 ssltss1 33331 . . . 4 (𝐴 <<s 𝐵𝐴 No )
98adantr 484 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 No )
10 ssltss2 33332 . . . . 5 (𝐴 <<s 𝐵𝐵 No )
1110adantr 484 . . . 4 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐵 No )
12 ssltss2 33332 . . . . 5 (𝐴 <<s 𝐶𝐶 No )
1312adantl 485 . . . 4 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐶 No )
1411, 13unssd 4137 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐵𝐶) ⊆ No )
15 ssltsep 33333 . . . . 5 (𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
1615adantr 484 . . . 4 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
17 ssltsep 33333 . . . . 5 (𝐴 <<s 𝐶 → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)
1817adantl 485 . . . 4 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)
19 ralunb 4142 . . . . . 6 (∀𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦 ↔ (∀𝑦𝐵 𝑥 <s 𝑦 ∧ ∀𝑦𝐶 𝑥 <s 𝑦))
2019ralbii 3157 . . . . 5 (∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦 ↔ ∀𝑥𝐴 (∀𝑦𝐵 𝑥 <s 𝑦 ∧ ∀𝑦𝐶 𝑥 <s 𝑦))
21 r19.26 3162 . . . . 5 (∀𝑥𝐴 (∀𝑦𝐵 𝑥 <s 𝑦 ∧ ∀𝑦𝐶 𝑥 <s 𝑦) ↔ (∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦 ∧ ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦))
2220, 21bitri 278 . . . 4 (∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦 ↔ (∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦 ∧ ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦))
2316, 18, 22sylanbrc 586 . . 3 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → ∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦)
249, 14, 233jca 1125 . 2 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → (𝐴 No ∧ (𝐵𝐶) ⊆ No ∧ ∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦))
25 brsslt 33328 . 2 (𝐴 <<s (𝐵𝐶) ↔ ((𝐴 ∈ V ∧ (𝐵𝐶) ∈ V) ∧ (𝐴 No ∧ (𝐵𝐶) ⊆ No ∧ ∀𝑥𝐴𝑦 ∈ (𝐵𝐶)𝑥 <s 𝑦)))
267, 24, 25sylanbrc 586 1 ((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 <<s (𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2114  ∀wral 3130  Vcvv 3469   ∪ cun 3906   ⊆ wss 3908   class class class wbr 5042   No csur 33221
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