Theorem sltsun2 27859

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 27833	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
2	1	adantr 484	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 ∈ V)
3		sltsex2 27834	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
4	3	adantr 484	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐵 ∈ V)
5		sltsex2 27834	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐶 ∈ V)
6	5	adantl 485	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐶 ∈ V)
7	4, 6	unexd 7733	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝐵 ∪ 𝐶) ∈ V)
8		sltsss1 27835	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
9	8	adantr 484	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 ⊆ No )
10		sltsss2 27836	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
11	10	adantr 484	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐵 ⊆ No )
12		sltsss2 27836	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐶 ⊆ No )
13	12	adantl 485	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐶 ⊆ No )
14	11, 13	unssd 4144	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝐵 ∪ 𝐶) ⊆ No )
15		elun 4106	. . . 4 ⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶))
16		sltssepc 27841	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 <s 𝑦)
17	16	3exp 1131	. . . . . . 7 ⊢ (𝐴 <<s 𝐵 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑥 <s 𝑦)))
18	17	adantr 484	. . . . . 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 27841	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐶 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → 𝑥 <s 𝑦)
21	20	3exp 1131	. . . . . . 7 ⊢ (𝐴 <<s 𝐶 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐶 → 𝑥 <s 𝑦)))
22	21	adantl 485	. . . . . 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 868	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶) → ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 <s 𝑦)))
25	15, 24	sylbi 219	. . 3 ⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) → ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 <s 𝑦)))
26	25	3imp231 1124	. 2 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 ∪ 𝐶)) → 𝑥 <s 𝑦)
27	2, 7, 9, 14, 26	sltsd 27838	1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 <<s (𝐵 ∪ 𝐶))

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 858   ∈ wcel 2141  Vcvv 3453   ∪ cun 3902   ⊆ wss 3904   class class class wbr 5099   No csur 27681   <s clts 27682   <<s cslts 27827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-xp 5651  df-slts 27828

This theorem is referenced by:  cutsun12  27860