Theorem ssltun1 27691

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

Assertion

Ref	Expression
ssltun1	⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝐴 ∪ 𝐵) <<s 𝐶)

Proof of Theorem ssltun1

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

Step	Hyp	Ref	Expression
1		ssltex1 27669	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐴 ∈ V)
2	1	adantr 480	. . 3 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐴 ∈ V)
3		ssltex1 27669	. . . 4 ⊢ (𝐵 <<s 𝐶 → 𝐵 ∈ V)
4	3	adantl 481	. . 3 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐵 ∈ V)
5	2, 4	unexd 7737	. 2 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝐴 ∪ 𝐵) ∈ V)
6		ssltex2 27670	. . 3 ⊢ (𝐴 <<s 𝐶 → 𝐶 ∈ V)
7	6	adantr 480	. 2 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐶 ∈ V)
8		ssltss1 27671	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐴 ⊆ No )
9	8	adantr 480	. . 3 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐴 ⊆ No )
10		ssltss1 27671	. . . 4 ⊢ (𝐵 <<s 𝐶 → 𝐵 ⊆ No )
11	10	adantl 481	. . 3 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐵 ⊆ No )
12	9, 11	unssd 4181	. 2 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝐴 ∪ 𝐵) ⊆ No )
13		ssltss2 27672	. . 3 ⊢ (𝐴 <<s 𝐶 → 𝐶 ⊆ No )
14	13	adantr 480	. 2 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐶 ⊆ No )
15		elun 4143	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
16		ssltsepc 27676	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐶 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → 𝑥 <s 𝑦)
17	16	3exp 1116	. . . . . . 7 ⊢ (𝐴 <<s 𝐶 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
18	17	adantr 480	. . . . . 6 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
19	18	com12 32	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
20		ssltsepc 27676	. . . . . . . 8 ⊢ ((𝐵 <<s 𝐶 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝑥 <s 𝑦)
21	20	3exp 1116	. . . . . . 7 ⊢ (𝐵 <<s 𝐶 → (𝑥 ∈ 𝐵 → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
22	21	adantl 481	. . . . . 6 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝑥 ∈ 𝐵 → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
23	22	com12 32	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
24	19, 23	jaoi 854	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
25	15, 24	sylbi 216	. . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) → ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
26	25	3imp21 1111	. 2 ⊢ (((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) ∧ 𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶) → 𝑥 <s 𝑦)
27	5, 7, 12, 14, 26	ssltd 27674	1 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝐴 ∪ 𝐵) <<s 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