Theorem ssltun2 31916

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 31901	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
2	1	adantr 481	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 ∈ V)
3		ssltex2 31902	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
4		ssltex2 31902	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐶 ∈ V)
5		unexg 6959	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐵 ∪ 𝐶) ∈ V)
6	3, 4, 5	syl2an 494	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝐵 ∪ 𝐶) ∈ V)
7	2, 6	jca 554	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝐴 ∈ V ∧ (𝐵 ∪ 𝐶) ∈ V))
8		ssltss1 31903	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
9	8	adantr 481	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 ⊆ No )
10		ssltss2 31904	. . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
11	10	adantr 481	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐵 ⊆ No )
12		ssltss2 31904	. . . . 5 ⊢ (𝐴 <<s 𝐶 → 𝐶 ⊆ No )
13	12	adantl 482	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐶 ⊆ No )
14	11, 13	unssd 3789	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝐵 ∪ 𝐶) ⊆ No )
15		ssltsep 31905	. . . . 5 ⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)
16	15	adantr 481	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)
17		ssltsep 31905	. . . . 5 ⊢ (𝐴 <<s 𝐶 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)
18	17	adantl 482	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)
19		ralunb 3794	. . . . . 6 ⊢ (∀𝑦 ∈ (𝐵 ∪ 𝐶)𝑥 <s 𝑦 ↔ (∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ∧ ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦))
20	19	ralbii 2980	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐵 ∪ 𝐶)𝑥 <s 𝑦 ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ∧ ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦))
21		r19.26 3064	. . . . 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 264	. . . 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 698	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐵 ∪ 𝐶)𝑥 <s 𝑦)
24	9, 14, 23	3jca 1242	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → (𝐴 ⊆ No ∧ (𝐵 ∪ 𝐶) ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐵 ∪ 𝐶)𝑥 <s 𝑦))
25		brsslt 31900	. 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 698	1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 <<s (𝐵 ∪ 𝐶))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∪ cun 3572   ⊆ wss 3574   class class class wbr 4653   No csur 31793   <s cslt 31794   <<s csslt 31896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-sslt 31897

This theorem is referenced by:  scutun12  31917