Theorem ssltun2 32797

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 32782	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
2	1	adantr 473	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 ∈ V)
3		ssltex2 32783	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
4		ssltex2 32783	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐶 ∈ V)
5		unexg 7289	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐵 ∪ 𝐶) ∈ V)
6	3, 4, 5	syl2an 586	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝐵 ∪ 𝐶) ∈ V)
7	2, 6	jca 504	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝐴 ∈ V ∧ (𝐵 ∪ 𝐶) ∈ V))
8		ssltss1 32784	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
9	8	adantr 473	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 ⊆ No )
10		ssltss2 32785	. . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
11	10	adantr 473	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐵 ⊆ No )
12		ssltss2 32785	. . . . 5 ⊢ (𝐴 <<s 𝐶 → 𝐶 ⊆ No )
13	12	adantl 474	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐶 ⊆ No )
14	11, 13	unssd 4050	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝐵 ∪ 𝐶) ⊆ No )
15		ssltsep 32786	. . . . 5 ⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)
16	15	adantr 473	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)
17		ssltsep 32786	. . . . 5 ⊢ (𝐴 <<s 𝐶 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)
18	17	adantl 474	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)
19		ralunb 4055	. . . . . 6 ⊢ (∀𝑦 ∈ (𝐵 ∪ 𝐶)𝑥 <s 𝑦 ↔ (∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ∧ ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦))
20	19	ralbii 3115	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐵 ∪ 𝐶)𝑥 <s 𝑦 ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ∧ ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦))
21		r19.26 3120	. . . . 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 267	. . . 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 575	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐵 ∪ 𝐶)𝑥 <s 𝑦)
24	9, 14, 23	3jca 1108	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝐴 ⊆ No ∧ (𝐵 ∪ 𝐶) ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐵 ∪ 𝐶)𝑥 <s 𝑦))
25		brsslt 32781	. 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 575	1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 <<s (𝐵 ∪ 𝐶))

Syntax hints:   → wi 4   ∧ wa 387   ∧ w3a 1068   ∈ wcel 2050  ∀wral 3088  Vcvv 3415   ∪ cun 3827   ⊆ wss 3829   class class class wbr 4929   No csur 32674   <s cslt 32675   <<s csslt 32777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186  ax-un 7279

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-xp 5413  df-sslt 32778

This theorem is referenced by:  scutun12  32798