Theorem sltsun1 28179

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

Assertion

Ref	Expression
sltsun1	⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝐴 ∪ 𝐵) <<s 𝐶)

Proof of Theorem sltsun1

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

Step	Hyp	Ref	Expression
1		sltsex1 28154	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐴 ∈ V)
2	1	adantr 486	. . 3 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐴 ∈ V)
3		sltsex1 28154	. . . 4 ⊢ (𝐵 <<s 𝐶 → 𝐵 ∈ V)
4	3	adantl 487	. . 3 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐵 ∈ V)
5	2, 4	unexd 7768	. 2 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝐴 ∪ 𝐵) ∈ V)
6		sltsex2 28155	. . 3 ⊢ (𝐴 <<s 𝐶 → 𝐶 ∈ V)
7	6	adantr 486	. 2 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐶 ∈ V)
8		sltsss1 28156	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐴 ⊆ No)
9	8	adantr 486	. . 3 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐴 ⊆ No)
10		sltsss1 28156	. . . 4 ⊢ (𝐵 <<s 𝐶 → 𝐵 ⊆ No)
11	10	adantl 487	. . 3 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐵 ⊆ No)
12	9, 11	unssd 4190	. 2 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝐴 ∪ 𝐵) ⊆ No)
13		sltsss2 28157	. . 3 ⊢ (𝐴 <<s 𝐶 → 𝐶 ⊆ No)
14	13	adantr 486	. 2 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐶 ⊆ No)
15		elun 4152	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
16		sltssepc 28162	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐶 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → 𝑥 <s 𝑦)
17	16	3exp 1137	. . . . . . 7 ⊢ (𝐴 <<s 𝐶 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
18	17	adantr 486	. . . . . 6 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
19	18	com12 33	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
20		sltssepc 28162	. . . . . . . 8 ⊢ ((𝐵 <<s 𝐶 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝑥 <s 𝑦)
21	20	3exp 1137	. . . . . . 7 ⊢ (𝐵 <<s 𝐶 → (𝑥 ∈ 𝐵 → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
22	21	adantl 487	. . . . . 6 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝑥 ∈ 𝐵 → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
23	22	com12 33	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
24	19, 23	jaoi 871	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
25	15, 24	sylbi 220	. . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) → ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
26	25	3imp21 1131	. 2 ⊢ (((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) ∧ 𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶) → 𝑥 <s 𝑦)
27	5, 7, 12, 14, 26	sltsd 28159	1 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝐴 ∪ 𝐵) <<s 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895   ∪ cun 4149   class class class wbr 5102  Nocsur 28002   <s clts 28003   <<s cslts 28148

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-slts 28149

This theorem is used by:  cutsun12  28181