Theorem sltsun1 27962

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 27937	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐴 ∈ V)
2	1	adantr 485	. . 3 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐴 ∈ V)
3		sltsex1 27937	. . . 4 ⊢ (𝐵 <<s 𝐶 → 𝐵 ∈ V)
4	3	adantl 486	. . 3 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐵 ∈ V)
5	2, 4	unexd 7754	. 2 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝐴 ∪ 𝐵) ∈ V)
6		sltsex2 27938	. . 3 ⊢ (𝐴 <<s 𝐶 → 𝐶 ∈ V)
7	6	adantr 485	. 2 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐶 ∈ V)
8		sltsss1 27939	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐴 ⊆ No )
9	8	adantr 485	. . 3 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐴 ⊆ No )
10		sltsss1 27939	. . . 4 ⊢ (𝐵 <<s 𝐶 → 𝐵 ⊆ No )
11	10	adantl 486	. . 3 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐵 ⊆ No )
12	9, 11	unssd 4146	. 2 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝐴 ∪ 𝐵) ⊆ No )
13		sltsss2 27940	. . 3 ⊢ (𝐴 <<s 𝐶 → 𝐶 ⊆ No )
14	13	adantr 485	. 2 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐶 ⊆ No )
15		elun 4108	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
16		sltssepc 27945	. . . . . . . 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 485	. . . . . 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 27945	. . . . . . . 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 486	. . . . . 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 870	. . . 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 27942	1 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝐴 ∪ 𝐵) <<s 𝐶)

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   ∈ wcel 2143  Vcvv 3455   ∪ cun 3904   ⊆ wss 3906   class class class wbr 5110   No csur 27785   <s clts 27786   <<s cslts 27931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-xp 5669  df-slts 27932

This theorem is referenced by:  cutsun12  27964