Theorem ssltun1 27871

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 27849	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐴 ∈ V)
2	1	adantr 480	. . 3 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐴 ∈ V)
3		ssltex1 27849	. . . 4 ⊢ (𝐵 <<s 𝐶 → 𝐵 ∈ V)
4	3	adantl 481	. . 3 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐵 ∈ V)
5	2, 4	unexd 7789	. 2 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝐴 ∪ 𝐵) ∈ V)
6		ssltex2 27850	. . 3 ⊢ (𝐴 <<s 𝐶 → 𝐶 ∈ V)
7	6	adantr 480	. 2 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐶 ∈ V)
8		ssltss1 27851	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐴 ⊆ No )
9	8	adantr 480	. . 3 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐴 ⊆ No )
10		ssltss1 27851	. . . 4 ⊢ (𝐵 <<s 𝐶 → 𝐵 ⊆ No )
11	10	adantl 481	. . 3 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐵 ⊆ No )
12	9, 11	unssd 4215	. 2 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝐴 ∪ 𝐵) ⊆ No )
13		ssltss2 27852	. . 3 ⊢ (𝐴 <<s 𝐶 → 𝐶 ⊆ No )
14	13	adantr 480	. 2 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐶 ⊆ No )
15		elun 4176	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
16		ssltsepc 27856	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐶 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → 𝑥 <s 𝑦)
17	16	3exp 1119	. . . . . . 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 27856	. . . . . . . 8 ⊢ ((𝐵 <<s 𝐶 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝑥 <s 𝑦)
21	20	3exp 1119	. . . . . . 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 856	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
25	15, 24	sylbi 217	. . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) → ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
26	25	3imp21 1114	. 2 ⊢ (((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) ∧ 𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶) → 𝑥 <s 𝑦)
27	5, 7, 12, 14, 26	ssltd 27854	1 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝐴 ∪ 𝐵) <<s 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 846   ∈ wcel 2108  Vcvv 3488   ∪ cun 3974   ⊆ wss 3976   class class class wbr 5166   No csur 27702   <s cslt 27703   <<s csslt 27843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-sslt 27844

This theorem is referenced by:  scutun12  27873