Theorem sltsun2 28157

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 28131	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
2	1	adantr 486	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 ∈ V)
3		sltsex2 28132	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
4	3	adantr 486	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐵 ∈ V)
5		sltsex2 28132	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐶 ∈ V)
6	5	adantl 487	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐶 ∈ V)
7	4, 6	unexd 7757	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝐵 ∪ 𝐶) ∈ V)
8		sltsss1 28133	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
9	8	adantr 486	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 ⊆ No )
10		sltsss2 28134	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
11	10	adantr 486	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐵 ⊆ No )
12		sltsss2 28134	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐶 ⊆ No )
13	12	adantl 487	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐶 ⊆ No )
14	11, 13	unssd 4138	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝐵 ∪ 𝐶) ⊆ No )
15		elun 4100	. . . 4 ⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶))
16		sltssepc 28139	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 <s 𝑦)
17	16	3exp 1137	. . . . . . 7 ⊢ (𝐴 <<s 𝐵 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑥 <s 𝑦)))
18	17	adantr 486	. . . . . 6 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑥 <s 𝑦)))
19	18	com3r 88	. . . . 5 ⊢ (𝑦 ∈ 𝐵 → ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 <s 𝑦)))
20		sltssepc 28139	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐶 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → 𝑥 <s 𝑦)
21	20	3exp 1137	. . . . . . 7 ⊢ (𝐴 <<s 𝐶 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
22	21	adantl 487	. . . . . 6 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
23	22	com3r 88	. . . . 5 ⊢ (𝑦 ∈ 𝐶 → ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 <s 𝑦)))
24	19, 23	jaoi 871	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶) → ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 <s 𝑦)))
25	15, 24	sylbi 220	. . 3 ⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) → ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 <s 𝑦)))
26	25	3imp231 1130	. 2 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 ∪ 𝐶)) → 𝑥 <s 𝑦)
27	2, 7, 9, 14, 26	sltsd 28136	1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 <<s (𝐵 ∪ 𝐶))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∈ wcel 2145  Vcvv 3451   ∪ cun 3897   ⊆ wss 3899   class class class wbr 5103   No csur 27979   <s clts 27980   <<s cslts 28125

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5657  df-slts 28126

This theorem is used by:  cutsun12  28158