Theorem ssltun2 27692

Description: Union law for surreal set less-than. (Contributed by Scott Fenton, 9-Dec-2021.)

Assertion

Ref	Expression
ssltun2	⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 <<s (𝐵 ∪ 𝐶))

Proof of Theorem ssltun2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssltex1 27669	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
2	1	adantr 480	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 ∈ V)
3		ssltex2 27670	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
4	3	adantr 480	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐵 ∈ V)
5		ssltex2 27670	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐶 ∈ V)
6	5	adantl 481	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐶 ∈ V)
7	4, 6	unexd 7737	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝐵 ∪ 𝐶) ∈ V)
8		ssltss1 27671	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
9	8	adantr 480	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 ⊆ No )
10		ssltss2 27672	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
11	10	adantr 480	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐵 ⊆ No )
12		ssltss2 27672	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐶 ⊆ No )
13	12	adantl 481	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐶 ⊆ No )
14	11, 13	unssd 4181	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝐵 ∪ 𝐶) ⊆ No )
15		elun 4143	. . . 4 ⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶))
16		ssltsepc 27676	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 <s 𝑦)
17	16	3exp 1116	. . . . . . 7 ⊢ (𝐴 <<s 𝐵 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑥 <s 𝑦)))
18	17	adantr 480	. . . . . 6 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑥 <s 𝑦)))
19	18	com3r 87	. . . . 5 ⊢ (𝑦 ∈ 𝐵 → ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 <s 𝑦)))
20		ssltsepc 27676	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐶 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → 𝑥 <s 𝑦)
21	20	3exp 1116	. . . . . . 7 ⊢ (𝐴 <<s 𝐶 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
22	21	adantl 481	. . . . . 6 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
23	22	com3r 87	. . . . 5 ⊢ (𝑦 ∈ 𝐶 → ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 <s 𝑦)))
24	19, 23	jaoi 854	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶) → ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 <s 𝑦)))
25	15, 24	sylbi 216	. . 3 ⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) → ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 <s 𝑦)))
26	25	3imp231 1110	. 2 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 ∪ 𝐶)) → 𝑥 <s 𝑦)
27	2, 7, 9, 14, 26	ssltd 27674	1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 <<s (𝐵 ∪ 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 844   ∈ wcel 2098  Vcvv 3468   ∪ cun 3941   ⊆ wss 3943   class class class wbr 5141   No csur 27523   <s cslt 27524   <<s csslt 27663

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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-sslt 27664

This theorem is referenced by:  scutun12  27693