Theorem sltsun2 27781

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

Assertion

Ref	Expression
sltsun2	⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 <<s (𝐵 ∪ 𝐶))

Proof of Theorem sltsun2

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

Step	Hyp	Ref	Expression
1		sltsex1 27755	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
2	1	adantr 480	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 ∈ V)
3		sltsex2 27756	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
4	3	adantr 480	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐵 ∈ V)
5		sltsex2 27756	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐶 ∈ V)
6	5	adantl 481	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐶 ∈ V)
7	4, 6	unexd 7708	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝐵 ∪ 𝐶) ∈ V)
8		sltsss1 27757	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
9	8	adantr 480	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 ⊆ No )
10		sltsss2 27758	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
11	10	adantr 480	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐵 ⊆ No )
12		sltsss2 27758	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐶 ⊆ No )
13	12	adantl 481	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐶 ⊆ No )
14	11, 13	unssd 4132	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝐵 ∪ 𝐶) ⊆ No )
15		elun 4093	. . . 4 ⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶))
16		sltssepc 27763	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 <s 𝑦)
17	16	3exp 1120	. . . . . . 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		sltssepc 27763	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐶 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → 𝑥 <s 𝑦)
21	20	3exp 1120	. . . . . . 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 858	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶) → ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 <s 𝑦)))
25	15, 24	sylbi 217	. . 3 ⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) → ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 <s 𝑦)))
26	25	3imp231 1113	. 2 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 ∪ 𝐶)) → 𝑥 <s 𝑦)
27	2, 7, 9, 14, 26	sltsd 27760	1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 <<s (𝐵 ∪ 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   ∈ wcel 2114  Vcvv 3429   ∪ cun 3887   ⊆ wss 3889   class class class wbr 5085   No csur 27603   <s clts 27604   <<s cslts 27749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-xp 5637  df-slts 27750

This theorem is referenced by:  cutsun12  27782