Theorem clsk1indlem3 45043

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 4525	. . . . . 6 ⊢ (𝑥 ∈ if((𝑠 ∪ 𝑡) = {∅}, {∅, 1o}, (𝑠 ∪ 𝑡)) ↔ (((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))))
2		uneq12 4162	. . . . . . . . . . 11 ⊢ ((𝑠 = {∅} ∧ 𝑡 = {∅}) → (𝑠 ∪ 𝑡) = ({∅} ∪ {∅}))
3		unidm 4156	. . . . . . . . . . 11 ⊢ ({∅} ∪ {∅}) = {∅}
4	2, 3	eqtrdi 2811	. . . . . . . . . 10 ⊢ ((𝑠 = {∅} ∧ 𝑡 = {∅}) → (𝑠 ∪ 𝑡) = {∅})
5		an3 672	. . . . . . . . . . . . . 14 ⊢ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∧ ((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o})) → (𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}))
6	5	orcd 887	. . . . . . . . . . . . 13 ⊢ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∧ ((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o})) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)))
7	6	orcd 887	. . . . . . . . . . . 12 ⊢ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∧ ((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o})) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
8	7	ex 418	. . . . . . . . . . 11 ⊢ ((𝑠 = {∅} ∧ 𝑡 = {∅}) → (((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))))
9		pm2.24 125	. . . . . . . . . . . 12 ⊢ ((𝑠 ∪ 𝑡) = {∅} → (¬ (𝑠 ∪ 𝑡) = {∅} → (𝑥 ∈ (𝑠 ∪ 𝑡) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
10	9	impd 416	. . . . . . . . . . 11 ⊢ ((𝑠 ∪ 𝑡) = {∅} → ((¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡)) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))))
11	8, 10	jaao 969	. . . . . . . . . 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 700	. . . . . . . . 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 45025	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∪ 𝑡) = {∅} ↔ ((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅) ∨ (𝑠 = ∅ ∧ 𝑡 = {∅})))
15		df-3or 1104	. . . . . . . . . . . . 13 ⊢ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅) ∨ (𝑠 = ∅ ∧ 𝑡 = {∅})) ↔ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅)) ∨ (𝑠 = ∅ ∧ 𝑡 = {∅})))
16	14, 15	bitri 278	. . . . . . . . . . . 12 ⊢ ((𝑠 ∪ 𝑡) = {∅} ↔ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅)) ∨ (𝑠 = ∅ ∧ 𝑡 = {∅})))
17		pm2.21 124	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑠 = {∅} → (𝑠 = {∅} → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
18	17	adantrd 497	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑠 = {∅} → ((𝑠 = {∅} ∧ 𝑡 = {∅}) → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
19	17	adantrd 497	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑠 = {∅} → ((𝑠 = {∅} ∧ 𝑡 = ∅) → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
20	18, 19	jaod 873	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑠 = {∅} → (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅)) → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
21	20	adantr 486	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅)) → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
22		pm2.21 124	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑡 = {∅} → (𝑡 = {∅} → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
23	22	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → (𝑡 = {∅} → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
24	23	adantld 496	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = ∅ ∧ 𝑡 = {∅}) → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
25	21, 24	jaod 873	. . . . . . . . . . . 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	biimtrid 245	. . . . . . . . . . 11 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 ∪ 𝑡) = {∅} → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
27	26	impd 416	. . . . . . . . . 10 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → (((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))))
28		elun 4152	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑠 ∪ 𝑡) ↔ (𝑥 ∈ 𝑠 ∨ 𝑥 ∈ 𝑡))
29	28	bilani 510	. . . . . . . . . . . . . 14 ⊢ ((¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡)) → (𝑥 ∈ 𝑠 ∨ 𝑥 ∈ 𝑡))
30		andi 1025	. . . . . . . . . . . . . . 15 ⊢ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ (𝑥 ∈ 𝑠 ∨ 𝑥 ∈ 𝑡)) ↔ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥 ∈ 𝑠) ∨ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥 ∈ 𝑡)))
31		simpl 488	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ¬ 𝑠 = {∅})
32	31	anim1i 627	. . . . . . . . . . . . . . . 16 ⊢ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥 ∈ 𝑠) → (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠))
33		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ¬ 𝑡 = {∅})
34	33	anim1i 627	. . . . . . . . . . . . . . . 16 ⊢ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥 ∈ 𝑡) → (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))
35	32, 34	orim12i 922	. . . . . . . . . . . . . . 15 ⊢ ((((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥 ∈ 𝑠) ∨ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥 ∈ 𝑡)) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))
36	30, 35	sylbi 220	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ (𝑥 ∈ 𝑠 ∨ 𝑥 ∈ 𝑡)) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))
37	29, 36	sylan2 605	. . . . . . . . . . . . 13 ⊢ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))
38	37	olcd 888	. . . . . . . . . . . 12 ⊢ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o})) ∨ ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
39		or4 940	. . . . . . . . . . . 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))))
40	38, 39	sylib 221	. . . . . . . . . . 11 ⊢ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
41	40	ex 418	. . . . . . . . . 10 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡)) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))))
42	27, 41	jaod 873	. . . . . . . . 9 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))))
43	42	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))))))
44	13, 43	jaod 873	. . . . . . 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))))))
45		orc 881	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))
46	45	expcom 419	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {∅, 1o} → (𝑠 = {∅} → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
47	46	adantrd 497	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {∅, 1o} → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
48	47	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
49		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝑠 ∧ 𝑠 = {∅}) → 𝑠 = {∅})
50		id 23	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = {∅} → 𝑠 = {∅})
51		snsspr1 4774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {∅} ⊆ {∅, 1o}
52	50, 51	eqsstrdi 3952	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = {∅} → 𝑠 ⊆ {∅, 1o})
53	52	sseld 3907	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = {∅} → (𝑥 ∈ 𝑠 → 𝑥 ∈ {∅, 1o}))
54	53	impcom 413	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝑠 ∧ 𝑠 = {∅}) → 𝑥 ∈ {∅, 1o})
55	49, 54	jca 521	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝑠 ∧ 𝑠 = {∅}) → (𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}))
56	55	orcd 887	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑠 ∧ 𝑠 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))
57	56	ex 418	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑠 → (𝑠 = {∅} → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
58		olc 882	. . . . . . . . . . . . . . . 16 ⊢ ((¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))
59	58	expcom 419	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑡 → (¬ 𝑡 = {∅} → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
60	57, 59	jaoa 970	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑠 ∨ 𝑥 ∈ 𝑡) → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
61	28, 60	sylbi 220	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑠 ∪ 𝑡) → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
62	61	adantl 487	. . . . . . . . . . . 12 ⊢ ((¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡)) → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
63	48, 62	jaoi 871	. . . . . . . . . . 11 ⊢ ((((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
64		olc 882	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o})))
65	64	expcom 419	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {∅, 1o} → (𝑡 = {∅} → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
66	65	adantl 487	. . . . . . . . . . . . 13 ⊢ (((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) → (𝑡 = {∅} → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
67	66	adantrd 497	. . . . . . . . . . . 12 ⊢ (((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) → ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
68		id 23	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) → (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠))
69	68	ex 418	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑠 = {∅} → (𝑥 ∈ 𝑠 → (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)))
70	69	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → (𝑥 ∈ 𝑠 → (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)))
71		id 23	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = {∅} → 𝑡 = {∅})
72	71, 51	eqsstrdi 3952	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = {∅} → 𝑡 ⊆ {∅, 1o})
73	72	sseld 3907	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = {∅} → (𝑥 ∈ 𝑡 → 𝑥 ∈ {∅, 1o}))
74	73	anc2li 565	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = {∅} → (𝑥 ∈ 𝑡 → (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o})))
75	74	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → (𝑥 ∈ 𝑡 → (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o})))
76	70, 75	orim12d 979	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → ((𝑥 ∈ 𝑠 ∨ 𝑥 ∈ 𝑡) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
77	76	com12 33	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑠 ∨ 𝑥 ∈ 𝑡) → ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
78	28, 77	sylbi 220	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑠 ∪ 𝑡) → ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
79	78	adantl 487	. . . . . . . . . . . 12 ⊢ ((¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡)) → ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
80	67, 79	jaoi 871	. . . . . . . . . . 11 ⊢ ((((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
81	63, 80	orim12d 979	. . . . . . . . . 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})))))
82	81	com12 33	. . . . . . . . 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})))))
83		or42 941	. . . . . . . . 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))))
84	82, 83	imbitrdi 254	. . . . . . . 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)))))
85	84	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))))))
86		4exmid 1067	. . . . . . . 8 ⊢ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅})) ∨ ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∨ (𝑡 = {∅} ∧ ¬ 𝑠 = {∅})))
87	86	a1i 11	. . . . . . 7 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅})) ∨ ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∨ (𝑡 = {∅} ∧ ¬ 𝑠 = {∅}))))
88	44, 85, 87	mpjaod 874	. . . . . 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)))))
89	1, 88	biimtrid 245	. . . . 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)))))
90		elun 4152	. . . . . 6 ⊢ (𝑥 ∈ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∪ if(𝑡 = {∅}, {∅, 1o}, 𝑡)) ↔ (𝑥 ∈ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∨ 𝑥 ∈ if(𝑡 = {∅}, {∅, 1o}, 𝑡)))
91		elif 4525	. . . . . . 7 ⊢ (𝑥 ∈ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)))
92		elif 4525	. . . . . . 7 ⊢ (𝑥 ∈ if(𝑡 = {∅}, {∅, 1o}, 𝑡) ↔ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))
93	91, 92	orbi12i 928	. . . . . 6 ⊢ ((𝑥 ∈ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∨ 𝑥 ∈ if(𝑡 = {∅}, {∅, 1o}, 𝑡)) ↔ (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
94	90, 93	sylbbr 239	. . . . 5 ⊢ ((((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))) → 𝑥 ∈ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∪ if(𝑡 = {∅}, {∅, 1o}, 𝑡)))
95	89, 94	syl6 36	. . . 4 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → (𝑥 ∈ if((𝑠 ∪ 𝑡) = {∅}, {∅, 1o}, (𝑠 ∪ 𝑡)) → 𝑥 ∈ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∪ if(𝑡 = {∅}, {∅, 1o}, 𝑡))))
96	95	ssrdv 3914	. . 3 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → if((𝑠 ∪ 𝑡) = {∅}, {∅, 1o}, (𝑠 ∪ 𝑡)) ⊆ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∪ if(𝑡 = {∅}, {∅, 1o}, 𝑡)))
97		pwuncl 7784	. . . 4 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → (𝑠 ∪ 𝑡) ∈ 𝒫 3o)
98		eqeq1 2764	. . . . . 6 ⊢ (𝑟 = (𝑠 ∪ 𝑡) → (𝑟 = {∅} ↔ (𝑠 ∪ 𝑡) = {∅}))
99		id 23	. . . . . 6 ⊢ (𝑟 = (𝑠 ∪ 𝑡) → 𝑟 = (𝑠 ∪ 𝑡))
100	98, 99	ifbieq2d 4508	. . . . 5 ⊢ (𝑟 = (𝑠 ∪ 𝑡) → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if((𝑠 ∪ 𝑡) = {∅}, {∅, 1o}, (𝑠 ∪ 𝑡)))
101		clsk1indlem.k	. . . . 5 ⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))
102		prex 5395	. . . . . 6 ⊢ {∅, 1o} ∈ V
103		vex 3454	. . . . . . 7 ⊢ 𝑠 ∈ V
104		vex 3454	. . . . . . 7 ⊢ 𝑡 ∈ V
105	103, 104	unex 7761	. . . . . 6 ⊢ (𝑠 ∪ 𝑡) ∈ V
106	102, 105	ifex 4532	. . . . 5 ⊢ if((𝑠 ∪ 𝑡) = {∅}, {∅, 1o}, (𝑠 ∪ 𝑡)) ∈ V
107	100, 101, 106	fvmpt 6993	. . . 4 ⊢ ((𝑠 ∪ 𝑡) ∈ 𝒫 3o → (𝐾‘(𝑠 ∪ 𝑡)) = if((𝑠 ∪ 𝑡) = {∅}, {∅, 1o}, (𝑠 ∪ 𝑡)))
108	97, 107	syl 18	. . 3 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → (𝐾‘(𝑠 ∪ 𝑡)) = if((𝑠 ∪ 𝑡) = {∅}, {∅, 1o}, (𝑠 ∪ 𝑡)))
109		eqeq1 2764	. . . . . . 7 ⊢ (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
110		id 23	. . . . . . 7 ⊢ (𝑟 = 𝑠 → 𝑟 = 𝑠)
111	109, 110	ifbieq2d 4508	. . . . . 6 ⊢ (𝑟 = 𝑠 → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
112	102, 103	ifex 4532	. . . . . 6 ⊢ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ V
113	111, 101, 112	fvmpt 6993	. . . . 5 ⊢ (𝑠 ∈ 𝒫 3o → (𝐾‘𝑠) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
114	113	adantr 486	. . . 4 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → (𝐾‘𝑠) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
115		eqeq1 2764	. . . . . . 7 ⊢ (𝑟 = 𝑡 → (𝑟 = {∅} ↔ 𝑡 = {∅}))
116		id 23	. . . . . . 7 ⊢ (𝑟 = 𝑡 → 𝑟 = 𝑡)
117	115, 116	ifbieq2d 4508	. . . . . 6 ⊢ (𝑟 = 𝑡 → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(𝑡 = {∅}, {∅, 1o}, 𝑡))
118	102, 104	ifex 4532	. . . . . 6 ⊢ if(𝑡 = {∅}, {∅, 1o}, 𝑡) ∈ V
119	117, 101, 118	fvmpt 6993	. . . . 5 ⊢ (𝑡 ∈ 𝒫 3o → (𝐾‘𝑡) = if(𝑡 = {∅}, {∅, 1o}, 𝑡))
120	119	adantl 487	. . . 4 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → (𝐾‘𝑡) = if(𝑡 = {∅}, {∅, 1o}, 𝑡))
121	114, 120	uneq12d 4168	. . 3 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → ((𝐾‘𝑠) ∪ (𝐾‘𝑡)) = (if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∪ if(𝑡 = {∅}, {∅, 1o}, 𝑡)))
122	96, 108, 121	3sstr4d 3963	. 2 ⊢ ((𝑠 ∈ 𝒫 3o ∧ 𝑡 ∈ 𝒫 3o) → (𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡)))
123	122	rgen2 3202	1 ⊢ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895   ∪ cun 4149  ∅c0 4278  ifcif 4481  𝒫 cpw 4556  {csn 4583  {cpr 4585   ↦ cmpt 5185  ‘cfv 6538  1_oc1o 8469  3_oc3o 8471

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-xor 1541  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546

This theorem is used by:  clsk1independent  45046