Theorem clsk1indlem3 41542

Description: The ansatz closure function (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟)) has the K3 property of being sub-linear. (Contributed by RP, 6-Jul-2021.)

Hypothesis

Ref	Expression
clsk1indlem.k	⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))

Assertion

Ref	Expression
clsk1indlem3	⊢ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡))

Distinct variable group:   𝑠,𝑟,𝑡

Allowed substitution hints:   𝐾(𝑡,𝑠,𝑟)

Proof of Theorem clsk1indlem3

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elif 4499	. . . . . 6 ⊢ (𝑥 ∈ if((𝑠 ∪ 𝑡) = {∅}, {∅, 1o}, (𝑠 ∪ 𝑡)) ↔ (((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))))
2		uneq12 4088	. . . . . . . . . . 11 ⊢ ((𝑠 = {∅} ∧ 𝑡 = {∅}) → (𝑠 ∪ 𝑡) = ({∅} ∪ {∅}))
3		unidm 4082	. . . . . . . . . . 11 ⊢ ({∅} ∪ {∅}) = {∅}
4	2, 3	eqtrdi 2795	. . . . . . . . . 10 ⊢ ((𝑠 = {∅} ∧ 𝑡 = {∅}) → (𝑠 ∪ 𝑡) = {∅})
5		an3 655	. . . . . . . . . . . . . 14 ⊢ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∧ ((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o})) → (𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}))
6	5	orcd 869	. . . . . . . . . . . . 13 ⊢ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∧ ((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o})) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)))
7	6	orcd 869	. . . . . . . . . . . 12 ⊢ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∧ ((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o})) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
8	7	ex 412	. . . . . . . . . . 11 ⊢ ((𝑠 = {∅} ∧ 𝑡 = {∅}) → (((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))))
9		pm2.24 124	. . . . . . . . . . . 12 ⊢ ((𝑠 ∪ 𝑡) = {∅} → (¬ (𝑠 ∪ 𝑡) = {∅} → (𝑥 ∈ (𝑠 ∪ 𝑡) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
10	9	impd 410	. . . . . . . . . . 11 ⊢ ((𝑠 ∪ 𝑡) = {∅} → ((¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡)) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))))
11	8, 10	jaao 951	. . . . . . . . . 10 ⊢ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∧ (𝑠 ∪ 𝑡) = {∅}) → ((((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))))
12	4, 11	mpdan 683	. . . . . . . . 9 ⊢ ((𝑠 = {∅} ∧ 𝑡 = {∅}) → ((((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))))
13	12	a1i 11	. . . . . . . 8 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → ((𝑠 = {∅} ∧ 𝑡 = {∅}) → ((((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
14		uneqsn 41522	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∪ 𝑡) = {∅} ↔ ((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅) ∨ (𝑠 = ∅ ∧ 𝑡 = {∅})))
15		df-3or 1086	. . . . . . . . . . . . 13 ⊢ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅) ∨ (𝑠 = ∅ ∧ 𝑡 = {∅})) ↔ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅)) ∨ (𝑠 = ∅ ∧ 𝑡 = {∅})))
16	14, 15	bitri 274	. . . . . . . . . . . 12 ⊢ ((𝑠 ∪ 𝑡) = {∅} ↔ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅)) ∨ (𝑠 = ∅ ∧ 𝑡 = {∅})))
17		pm2.21 123	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑠 = {∅} → (𝑠 = {∅} → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
18	17	adantrd 491	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑠 = {∅} → ((𝑠 = {∅} ∧ 𝑡 = {∅}) → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
19	17	adantrd 491	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑠 = {∅} → ((𝑠 = {∅} ∧ 𝑡 = ∅) → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
20	18, 19	jaod 855	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑠 = {∅} → (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅)) → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
21	20	adantr 480	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅)) → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
22		pm2.21 123	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑡 = {∅} → (𝑡 = {∅} → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
23	22	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → (𝑡 = {∅} → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
24	23	adantld 490	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = ∅ ∧ 𝑡 = {∅}) → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
25	21, 24	jaod 855	. . . . . . . . . . . 12 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅)) ∨ (𝑠 = ∅ ∧ 𝑡 = {∅})) → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
26	16, 25	syl5bi 241	. . . . . . . . . . 11 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 ∪ 𝑡) = {∅} → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
27	26	impd 410	. . . . . . . . . 10 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → (((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))))
28		elun 4079	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑠 ∪ 𝑡) ↔ (𝑥 ∈ 𝑠 ∨ 𝑥 ∈ 𝑡))
29	28	biimpi 215	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑠 ∪ 𝑡) → (𝑥 ∈ 𝑠 ∨ 𝑥 ∈ 𝑡))
30	29	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡)) → (𝑥 ∈ 𝑠 ∨ 𝑥 ∈ 𝑡))
31		andi 1004	. . . . . . . . . . . . . . 15 ⊢ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ (𝑥 ∈ 𝑠 ∨ 𝑥 ∈ 𝑡)) ↔ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥 ∈ 𝑠) ∨ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥 ∈ 𝑡)))
32		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ¬ 𝑠 = {∅})
33	32	anim1i 614	. . . . . . . . . . . . . . . 16 ⊢ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥 ∈ 𝑠) → (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠))
34		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ¬ 𝑡 = {∅})
35	34	anim1i 614	. . . . . . . . . . . . . . . 16 ⊢ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥 ∈ 𝑡) → (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))
36	33, 35	orim12i 905	. . . . . . . . . . . . . . 15 ⊢ ((((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥 ∈ 𝑠) ∨ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥 ∈ 𝑡)) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))
37	31, 36	sylbi 216	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ (𝑥 ∈ 𝑠 ∨ 𝑥 ∈ 𝑡)) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))
38	30, 37	sylan2 592	. . . . . . . . . . . . 13 ⊢ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))
39	38	olcd 870	. . . . . . . . . . . 12 ⊢ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o})) ∨ ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
40		or4 923	. . . . . . . . . . . 12 ⊢ ((((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o})) ∨ ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))) ↔ (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
41	39, 40	sylib 217	. . . . . . . . . . 11 ⊢ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
42	41	ex 412	. . . . . . . . . 10 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡)) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))))
43	27, 42	jaod 855	. . . . . . . . 9 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))))
44	43	a1i 11	. . . . . . . 8 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
45	13, 44	jaod 855	. . . . . . 7 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅})) → ((((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
46		orc 863	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))
47	46	expcom 413	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {∅, 1o} → (𝑠 = {∅} → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
48	47	adantrd 491	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {∅, 1o} → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
49	48	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
50		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝑠 ∧ 𝑠 = {∅}) → 𝑠 = {∅})
51		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = {∅} → 𝑠 = {∅})
52		snsspr1 4744	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {∅} ⊆ {∅, 1o}
53	51, 52	eqsstrdi 3971	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = {∅} → 𝑠 ⊆ {∅, 1o})
54	53	sseld 3916	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = {∅} → (𝑥 ∈ 𝑠 → 𝑥 ∈ {∅, 1o}))
55	54	impcom 407	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝑠 ∧ 𝑠 = {∅}) → 𝑥 ∈ {∅, 1o})
56	50, 55	jca 511	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝑠 ∧ 𝑠 = {∅}) → (𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}))
57	56	orcd 869	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑠 ∧ 𝑠 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))
58	57	ex 412	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑠 → (𝑠 = {∅} → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
59		olc 864	. . . . . . . . . . . . . . . 16 ⊢ ((¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))
60	59	expcom 413	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑡 → (¬ 𝑡 = {∅} → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
61	58, 60	jaoa 952	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑠 ∨ 𝑥 ∈ 𝑡) → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
62	28, 61	sylbi 216	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑠 ∪ 𝑡) → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
63	62	adantl 481	. . . . . . . . . . . 12 ⊢ ((¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡)) → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
64	49, 63	jaoi 853	. . . . . . . . . . 11 ⊢ ((((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
65		olc 864	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o})))
66	65	expcom 413	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {∅, 1o} → (𝑡 = {∅} → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
67	66	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) → (𝑡 = {∅} → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
68	67	adantrd 491	. . . . . . . . . . . 12 ⊢ (((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) → ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
69		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) → (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠))
70	69	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑠 = {∅} → (𝑥 ∈ 𝑠 → (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)))
71	70	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → (𝑥 ∈ 𝑠 → (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)))
72		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = {∅} → 𝑡 = {∅})
73	72, 52	eqsstrdi 3971	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = {∅} → 𝑡 ⊆ {∅, 1o})
74	73	sseld 3916	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = {∅} → (𝑥 ∈ 𝑡 → 𝑥 ∈ {∅, 1o}))
75	74	anc2li 555	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = {∅} → (𝑥 ∈ 𝑡 → (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o})))
76	75	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → (𝑥 ∈ 𝑡 → (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o})))
77	71, 76	orim12d 961	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → ((𝑥 ∈ 𝑠 ∨ 𝑥 ∈ 𝑡) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
78	77	com12 32	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑠 ∨ 𝑥 ∈ 𝑡) → ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
79	28, 78	sylbi 216	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑠 ∪ 𝑡) → ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
80	79	adantl 481	. . . . . . . . . . . 12 ⊢ ((¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡)) → ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
81	68, 80	jaoi 853	. . . . . . . . . . 11 ⊢ ((((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
82	64, 81	orim12d 961	. . . . . . . . . 10 ⊢ ((((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → (((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∨ (𝑡 = {∅} ∧ ¬ 𝑠 = {∅})) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)) ∨ ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o})))))
83	82	com12 32	. . . . . . . . 9 ⊢ (((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∨ (𝑡 = {∅} ∧ ¬ 𝑠 = {∅})) → ((((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)) ∨ ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o})))))
84		or42 924	. . . . . . . . 9 ⊢ ((((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)) ∨ ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))) ↔ (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
85	83, 84	syl6ib 250	. . . . . . . 8 ⊢ (((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∨ (𝑡 = {∅} ∧ ¬ 𝑠 = {∅})) → ((((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))))
86	85	a1i 11	. . . . . . 7 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → (((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∨ (𝑡 = {∅} ∧ ¬ 𝑠 = {∅})) → ((((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
87		4exmid 1048	. . . . . . . 8 ⊢ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅})) ∨ ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∨ (𝑡 = {∅} ∧ ¬ 𝑠 = {∅})))
88	87	a1i 11	. . . . . . 7 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅})) ∨ ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∨ (𝑡 = {∅} ∧ ¬ 𝑠 = {∅}))))
89	45, 86, 88	mpjaod 856	. . . . . 6 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → ((((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))))
90	1, 89	syl5bi 241	. . . . 5 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → (𝑥 ∈ if((𝑠 ∪ 𝑡) = {∅}, {∅, 1o}, (𝑠 ∪ 𝑡)) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))))
91		elun 4079	. . . . . 6 ⊢ (𝑥 ∈ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∪ if(𝑡 = {∅}, {∅, 1o}, 𝑡)) ↔ (𝑥 ∈ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∨ 𝑥 ∈ if(𝑡 = {∅}, {∅, 1o}, 𝑡)))
92		elif 4499	. . . . . . 7 ⊢ (𝑥 ∈ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)))
93		elif 4499	. . . . . . 7 ⊢ (𝑥 ∈ if(𝑡 = {∅}, {∅, 1o}, 𝑡) ↔ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))
94	92, 93	orbi12i 911	. . . . . 6 ⊢ ((𝑥 ∈ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∨ 𝑥 ∈ if(𝑡 = {∅}, {∅, 1o}, 𝑡)) ↔ (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
95	91, 94	sylbbr 235	. . . . 5 ⊢ ((((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))) → 𝑥 ∈ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∪ if(𝑡 = {∅}, {∅, 1o}, 𝑡)))
96	90, 95	syl6 35	. . . 4 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → (𝑥 ∈ if((𝑠 ∪ 𝑡) = {∅}, {∅, 1o}, (𝑠 ∪ 𝑡)) → 𝑥 ∈ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∪ if(𝑡 = {∅}, {∅, 1o}, 𝑡))))
97	96	ssrdv 3923	. . 3 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → if((𝑠 ∪ 𝑡) = {∅}, {∅, 1o}, (𝑠 ∪ 𝑡)) ⊆ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∪ if(𝑡 = {∅}, {∅, 1o}, 𝑡)))
98		pwuncl 7598	. . . 4 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → (𝑠 ∪ 𝑡) ∈ 𝒫 3o)
99		eqeq1 2742	. . . . . 6 ⊢ (𝑟 = (𝑠 ∪ 𝑡) → (𝑟 = {∅} ↔ (𝑠 ∪ 𝑡) = {∅}))
100		id 22	. . . . . 6 ⊢ (𝑟 = (𝑠 ∪ 𝑡) → 𝑟 = (𝑠 ∪ 𝑡))
101	99, 100	ifbieq2d 4482	. . . . 5 ⊢ (𝑟 = (𝑠 ∪ 𝑡) → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if((𝑠 ∪ 𝑡) = {∅}, {∅, 1o}, (𝑠 ∪ 𝑡)))
102		clsk1indlem.k	. . . . 5 ⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))
103		prex 5350	. . . . . 6 ⊢ {∅, 1o} ∈ V
104		vex 3426	. . . . . . 7 ⊢ 𝑠 ∈ V
105		vex 3426	. . . . . . 7 ⊢ 𝑡 ∈ V
106	104, 105	unex 7574	. . . . . 6 ⊢ (𝑠 ∪ 𝑡) ∈ V
107	103, 106	ifex 4506	. . . . 5 ⊢ if((𝑠 ∪ 𝑡) = {∅}, {∅, 1o}, (𝑠 ∪ 𝑡)) ∈ V
108	101, 102, 107	fvmpt 6857	. . . 4 ⊢ ((𝑠 ∪ 𝑡) ∈ 𝒫 3o → (𝐾‘(𝑠 ∪ 𝑡)) = if((𝑠 ∪ 𝑡) = {∅}, {∅, 1o}, (𝑠 ∪ 𝑡)))
109	98, 108	syl 17	. . 3 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → (𝐾‘(𝑠 ∪ 𝑡)) = if((𝑠 ∪ 𝑡) = {∅}, {∅, 1o}, (𝑠 ∪ 𝑡)))
110		eqeq1 2742	. . . . . . 7 ⊢ (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
111		id 22	. . . . . . 7 ⊢ (𝑟 = 𝑠 → 𝑟 = 𝑠)
112	110, 111	ifbieq2d 4482	. . . . . 6 ⊢ (𝑟 = 𝑠 → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
113	103, 104	ifex 4506	. . . . . 6 ⊢ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ V
114	112, 102, 113	fvmpt 6857	. . . . 5 ⊢ (𝑠 ∈ 𝒫 3o → (𝐾‘𝑠) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
115	114	adantr 480	. . . 4 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → (𝐾‘𝑠) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
116		eqeq1 2742	. . . . . . 7 ⊢ (𝑟 = 𝑡 → (𝑟 = {∅} ↔ 𝑡 = {∅}))
117		id 22	. . . . . . 7 ⊢ (𝑟 = 𝑡 → 𝑟 = 𝑡)
118	116, 117	ifbieq2d 4482	. . . . . 6 ⊢ (𝑟 = 𝑡 → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(𝑡 = {∅}, {∅, 1o}, 𝑡))
119	103, 105	ifex 4506	. . . . . 6 ⊢ if(𝑡 = {∅}, {∅, 1o}, 𝑡) ∈ V
120	118, 102, 119	fvmpt 6857	. . . . 5 ⊢ (𝑡 ∈ 𝒫 3o → (𝐾‘𝑡) = if(𝑡 = {∅}, {∅, 1o}, 𝑡))
121	120	adantl 481	. . . 4 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → (𝐾‘𝑡) = if(𝑡 = {∅}, {∅, 1o}, 𝑡))
122	115, 121	uneq12d 4094	. . 3 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → ((𝐾‘𝑠) ∪ (𝐾‘𝑡)) = (if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∪ if(𝑡 = {∅}, {∅, 1o}, 𝑡)))
123	97, 109, 122	3sstr4d 3964	. 2 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → (𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡)))
124	123	rgen2 3126	1 ⊢ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 843   ∨ w3o 1084   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  ifcif 4456  𝒫 cpw 4530  {csn 4558  {cpr 4560   ↦ cmpt 5153  ‘cfv 6418  1_oc1o 8260  3_oc3o 8262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-xor 1504  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426

This theorem is referenced by:  clsk1independent  41545