Theorem ssltun2 33265

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

Assertion

Ref	Expression
ssltun2	⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 <<s (𝐵 ∪ 𝐶))

Proof of Theorem ssltun2

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

Step	Hyp	Ref	Expression
1		ssltex1 33250	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
2	1	adantr 483	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 ∈ V)
3		ssltex2 33251	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
4		ssltex2 33251	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐶 ∈ V)
5		unexg 7466	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐵 ∪ 𝐶) ∈ V)
6	3, 4, 5	syl2an 597	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝐵 ∪ 𝐶) ∈ V)
7	2, 6	jca 514	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝐴 ∈ V ∧ (𝐵 ∪ 𝐶) ∈ V))
8		ssltss1 33252	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
9	8	adantr 483	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 ⊆ No )
10		ssltss2 33253	. . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
11	10	adantr 483	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐵 ⊆ No )
12		ssltss2 33253	. . . . 5 ⊢ (𝐴 <<s 𝐶 → 𝐶 ⊆ No )
13	12	adantl 484	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐶 ⊆ No )
14	11, 13	unssd 4162	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝐵 ∪ 𝐶) ⊆ No )
15		ssltsep 33254	. . . . 5 ⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)
16	15	adantr 483	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)
17		ssltsep 33254	. . . . 5 ⊢ (𝐴 <<s 𝐶 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)
18	17	adantl 484	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)
19		ralunb 4167	. . . . . 6 ⊢ (∀𝑦 ∈ (𝐵 ∪ 𝐶)𝑥 <s 𝑦 ↔ (∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ∧ ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦))
20	19	ralbii 3165	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐵 ∪ 𝐶)𝑥 <s 𝑦 ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ∧ ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦))
21		r19.26 3170	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ∧ ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦))
22	20, 21	bitri 277	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐵 ∪ 𝐶)𝑥 <s 𝑦 ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦))
23	16, 18, 22	sylanbrc 585	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐵 ∪ 𝐶)𝑥 <s 𝑦)
24	9, 14, 23	3jca 1124	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝐴 ⊆ No ∧ (𝐵 ∪ 𝐶) ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐵 ∪ 𝐶)𝑥 <s 𝑦))
25		brsslt 33249	. 2 ⊢ (𝐴 <<s (𝐵 ∪ 𝐶) ↔ ((𝐴 ∈ V ∧ (𝐵 ∪ 𝐶) ∈ V) ∧ (𝐴 ⊆ No ∧ (𝐵 ∪ 𝐶) ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐵 ∪ 𝐶)𝑥 <s 𝑦)))
26	7, 24, 25	sylanbrc 585	1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 <<s (𝐵 ∪ 𝐶))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   ∈ wcel 2110  ∀wral 3138  Vcvv 3495   ∪ cun 3934   ⊆ wss 3936   class class class wbr 5059   No csur 33142   <s cslt 33143   <<s csslt 33245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-xp 5556  df-sslt 33246

This theorem is referenced by:  scutun12  33266