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Theorem ssltun1 33377
 Description: Union law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.)
Assertion
Ref Expression
ssltun1 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝐴𝐵) <<s 𝐶)

Proof of Theorem ssltun1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssltex1 33363 . . . 4 (𝐴 <<s 𝐶𝐴 ∈ V)
2 ssltex1 33363 . . . 4 (𝐵 <<s 𝐶𝐵 ∈ V)
3 unexg 7456 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
41, 2, 3syl2an 598 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝐴𝐵) ∈ V)
5 ssltex2 33364 . . . 4 (𝐴 <<s 𝐶𝐶 ∈ V)
65adantr 484 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐶 ∈ V)
74, 6jca 515 . 2 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → ((𝐴𝐵) ∈ V ∧ 𝐶 ∈ V))
8 ssltss1 33365 . . . . 5 (𝐴 <<s 𝐶𝐴 No )
98adantr 484 . . . 4 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐴 No )
10 ssltss1 33365 . . . . 5 (𝐵 <<s 𝐶𝐵 No )
1110adantl 485 . . . 4 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐵 No )
129, 11unssd 4116 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝐴𝐵) ⊆ No )
13 ssltss2 33366 . . . 4 (𝐵 <<s 𝐶𝐶 No )
1413adantl 485 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → 𝐶 No )
15 ssltsep 33367 . . . . 5 (𝐴 <<s 𝐶 → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)
1615adantr 484 . . . 4 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → ∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦)
17 ssltsep 33367 . . . . 5 (𝐵 <<s 𝐶 → ∀𝑥𝐵𝑦𝐶 𝑥 <s 𝑦)
1817adantl 485 . . . 4 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → ∀𝑥𝐵𝑦𝐶 𝑥 <s 𝑦)
19 ralunb 4121 . . . 4 (∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 𝑥 <s 𝑦 ↔ (∀𝑥𝐴𝑦𝐶 𝑥 <s 𝑦 ∧ ∀𝑥𝐵𝑦𝐶 𝑥 <s 𝑦))
2016, 18, 19sylanbrc 586 . . 3 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → ∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 𝑥 <s 𝑦)
2112, 14, 203jca 1125 . 2 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → ((𝐴𝐵) ⊆ No 𝐶 No ∧ ∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 𝑥 <s 𝑦))
22 brsslt 33362 . 2 ((𝐴𝐵) <<s 𝐶 ↔ (((𝐴𝐵) ∈ V ∧ 𝐶 ∈ V) ∧ ((𝐴𝐵) ⊆ No 𝐶 No ∧ ∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 𝑥 <s 𝑦)))
237, 21, 22sylanbrc 586 1 ((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝐴𝐵) <<s 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2112  ∀wral 3109  Vcvv 3444   ∪ cun 3882   ⊆ wss 3884   class class class wbr 5033   No csur 33255
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