Theorem sltsun2 27797

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 27771	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
2	1	adantr 480	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 ∈ V)
3		sltsex2 27772	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
4	3	adantr 480	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐵 ∈ V)
5		sltsex2 27772	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐶 ∈ V)
6	5	adantl 481	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐶 ∈ V)
7	4, 6	unexd 7709	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝐵 ∪ 𝐶) ∈ V)
8		sltsss1 27773	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
9	8	adantr 480	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 ⊆ No )
10		sltsss2 27774	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
11	10	adantr 480	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐵 ⊆ No )
12		sltsss2 27774	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐶 ⊆ No )
13	12	adantl 481	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐶 ⊆ No )
14	11, 13	unssd 4146	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝐵 ∪ 𝐶) ⊆ No )
15		elun 4107	. . . 4 ⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶))
16		sltssepc 27779	. . . . . . . 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 27779	. . . . . . . 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 27776	1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 <<s (𝐵 ∪ 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   ∈ wcel 2114  Vcvv 3442   ∪ cun 3901   ⊆ wss 3903   class class class wbr 5100   No csur 27619   <s clts 27620   <<s cslts 27765

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 2709  ax-sep 5243  ax-pr 5379  ax-un 7690

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 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5638  df-slts 27766

This theorem is referenced by:  cutsun12  27798