Theorem ssltun1 33269

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

Assertion

Ref	Expression
ssltun1	⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝐴 ∪ 𝐵) <<s 𝐶)

Proof of Theorem ssltun1

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

Step	Hyp	Ref	Expression
1		ssltex1 33255	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐴 ∈ V)
2		ssltex1 33255	. . . 4 ⊢ (𝐵 <<s 𝐶 → 𝐵 ∈ V)
3		unexg 7472	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
4	1, 2, 3	syl2an 597	. . 3 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝐴 ∪ 𝐵) ∈ V)
5		ssltex2 33256	. . . 4 ⊢ (𝐴 <<s 𝐶 → 𝐶 ∈ V)
6	5	adantr 483	. . 3 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐶 ∈ V)
7	4, 6	jca 514	. 2 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → ((𝐴 ∪ 𝐵) ∈ V ∧ 𝐶 ∈ V))
8		ssltss1 33257	. . . . 5 ⊢ (𝐴 <<s 𝐶 → 𝐴 ⊆ No )
9	8	adantr 483	. . . 4 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐴 ⊆ No )
10		ssltss1 33257	. . . . 5 ⊢ (𝐵 <<s 𝐶 → 𝐵 ⊆ No )
11	10	adantl 484	. . . 4 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐵 ⊆ No )
12	9, 11	unssd 4162	. . 3 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝐴 ∪ 𝐵) ⊆ No )
13		ssltss2 33258	. . . 4 ⊢ (𝐵 <<s 𝐶 → 𝐶 ⊆ No )
14	13	adantl 484	. . 3 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → 𝐶 ⊆ No )
15		ssltsep 33259	. . . . 5 ⊢ (𝐴 <<s 𝐶 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)
16	15	adantr 483	. . . 4 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)
17		ssltsep 33259	. . . . 5 ⊢ (𝐵 <<s 𝐶 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)
18	17	adantl 484	. . . 4 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)
19		ralunb 4167	. . . 4 ⊢ (∀𝑥 ∈ (𝐴 ∪ 𝐵)∀𝑦 ∈ 𝐶 𝑥 <s 𝑦 ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦))
20	16, 18, 19	sylanbrc 585	. . 3 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)
21	12, 14, 20	3jca 1124	. 2 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → ((𝐴 ∪ 𝐵) ⊆ No ∧ 𝐶 ⊆ No ∧ ∀𝑥 ∈ (𝐴 ∪ 𝐵)∀𝑦 ∈ 𝐶 𝑥 <s 𝑦))
22		brsslt 33254	. 2 ⊢ ((𝐴 ∪ 𝐵) <<s 𝐶 ↔ (((𝐴 ∪ 𝐵) ∈ V ∧ 𝐶 ∈ V) ∧ ((𝐴 ∪ 𝐵) ⊆ No ∧ 𝐶 ⊆ No ∧ ∀𝑥 ∈ (𝐴 ∪ 𝐵)∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)))
23	7, 21, 22	sylanbrc 585	1 ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝐴 ∪ 𝐵) <<s 𝐶)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   ∈ wcel 2114  ∀wral 3138  Vcvv 3494   ∪ cun 3934   ⊆ wss 3936   class class class wbr 5066   No csur 33147   <s cslt 33148   <<s csslt 33250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-xp 5561  df-sslt 33251

This theorem is referenced by:  scutun12  33271