Theorem clsk1indlem3 38341

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 4128	. . . . . 6 ⊢ (𝑥 ∈ if((𝑠 ∪ 𝑡) = {∅}, {∅, 1𝑜}, (𝑠 ∪ 𝑡)) ↔ (((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))))
2		uneq12 3762	. . . . . . . . . . 11 ⊢ ((𝑠 = {∅} ∧ 𝑡 = {∅}) → (𝑠 ∪ 𝑡) = ({∅} ∪ {∅}))
3		unidm 3756	. . . . . . . . . . 11 ⊢ ({∅} ∪ {∅}) = {∅}
4	2, 3	syl6eq 2672	. . . . . . . . . 10 ⊢ ((𝑠 = {∅} ∧ 𝑡 = {∅}) → (𝑠 ∪ 𝑡) = {∅})
5		an3 868	. . . . . . . . . . . . . 14 ⊢ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∧ ((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1𝑜})) → (𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}))
6	5	orcd 407	. . . . . . . . . . . . 13 ⊢ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∧ ((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1𝑜})) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)))
7	6	orcd 407	. . . . . . . . . . . 12 ⊢ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∧ ((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1𝑜})) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
8	7	ex 450	. . . . . . . . . . 11 ⊢ ((𝑠 = {∅} ∧ 𝑡 = {∅}) → (((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))))
9		pm2.24 121	. . . . . . . . . . . 12 ⊢ ((𝑠 ∪ 𝑡) = {∅} → (¬ (𝑠 ∪ 𝑡) = {∅} → (𝑥 ∈ (𝑠 ∪ 𝑡) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
10	9	impd 447	. . . . . . . . . . 11 ⊢ ((𝑠 ∪ 𝑡) = {∅} → ((¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡)) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))))
11	8, 10	jaao 531	. . . . . . . . . 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 702	. . . . . . . . 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 38321	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∪ 𝑡) = {∅} ↔ ((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅) ∨ (𝑠 = ∅ ∧ 𝑡 = {∅})))
15		df-3or 1038	. . . . . . . . . . . . 13 ⊢ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅) ∨ (𝑠 = ∅ ∧ 𝑡 = {∅})) ↔ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅)) ∨ (𝑠 = ∅ ∧ 𝑡 = {∅})))
16	14, 15	bitri 264	. . . . . . . . . . . 12 ⊢ ((𝑠 ∪ 𝑡) = {∅} ↔ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅)) ∨ (𝑠 = ∅ ∧ 𝑡 = {∅})))
17		pm2.21 120	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑠 = {∅} → (𝑠 = {∅} → (𝑥 ∈ {∅, 1𝑜} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
18	17	adantrd 484	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑠 = {∅} → ((𝑠 = {∅} ∧ 𝑡 = {∅}) → (𝑥 ∈ {∅, 1𝑜} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
19	17	adantrd 484	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑠 = {∅} → ((𝑠 = {∅} ∧ 𝑡 = ∅) → (𝑥 ∈ {∅, 1𝑜} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
20	18, 19	jaod 395	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑠 = {∅} → (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅)) → (𝑥 ∈ {∅, 1𝑜} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
21	20	adantr 481	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅)) → (𝑥 ∈ {∅, 1𝑜} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
22		pm2.21 120	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑡 = {∅} → (𝑡 = {∅} → (𝑥 ∈ {∅, 1𝑜} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
23	22	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → (𝑡 = {∅} → (𝑥 ∈ {∅, 1𝑜} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
24	23	adantld 483	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = ∅ ∧ 𝑡 = {∅}) → (𝑥 ∈ {∅, 1𝑜} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
25	21, 24	jaod 395	. . . . . . . . . . . 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 232	. . . . . . . . . . 11 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 ∪ 𝑡) = {∅} → (𝑥 ∈ {∅, 1𝑜} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))))
27	26	impd 447	. . . . . . . . . 10 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → (((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))))
28		elun 3753	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑠 ∪ 𝑡) ↔ (𝑥 ∈ 𝑠 ∨ 𝑥 ∈ 𝑡))
29	28	biimpi 206	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑠 ∪ 𝑡) → (𝑥 ∈ 𝑠 ∨ 𝑥 ∈ 𝑡))
30	29	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡)) → (𝑥 ∈ 𝑠 ∨ 𝑥 ∈ 𝑡))
31		andi 911	. . . . . . . . . . . . . . 15 ⊢ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ (𝑥 ∈ 𝑠 ∨ 𝑥 ∈ 𝑡)) ↔ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥 ∈ 𝑠) ∨ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥 ∈ 𝑡)))
32		simpl 473	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ¬ 𝑠 = {∅})
33	32	anim1i 592	. . . . . . . . . . . . . . . 16 ⊢ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥 ∈ 𝑠) → (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠))
34		simpr 477	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ¬ 𝑡 = {∅})
35	34	anim1i 592	. . . . . . . . . . . . . . . 16 ⊢ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥 ∈ 𝑡) → (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))
36	33, 35	orim12i 538	. . . . . . . . . . . . . . 15 ⊢ ((((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥 ∈ 𝑠) ∨ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥 ∈ 𝑡)) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))
37	31, 36	sylbi 207	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ (𝑥 ∈ 𝑠 ∨ 𝑥 ∈ 𝑡)) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))
38	30, 37	sylan2 491	. . . . . . . . . . . . 13 ⊢ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))
39	38	olcd 408	. . . . . . . . . . . 12 ⊢ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜})) ∨ ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
40		or4 550	. . . . . . . . . . . 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 208	. . . . . . . . . . 11 ⊢ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
42	41	ex 450	. . . . . . . . . 10 ⊢ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡)) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))))
43	27, 42	jaod 395	. . . . . . . . 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 395	. . . . . . 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 400	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))
47	46	expcom 451	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {∅, 1𝑜} → (𝑠 = {∅} → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
48	47	adantrd 484	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {∅, 1𝑜} → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
49	48	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
50		simpr 477	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝑠 ∧ 𝑠 = {∅}) → 𝑠 = {∅})
51		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = {∅} → 𝑠 = {∅})
52		snsspr1 4345	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {∅} ⊆ {∅, 1𝑜}
53	51, 52	syl6eqss 3655	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = {∅} → 𝑠 ⊆ {∅, 1𝑜})
54	53	sseld 3602	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = {∅} → (𝑥 ∈ 𝑠 → 𝑥 ∈ {∅, 1𝑜}))
55	54	impcom 446	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝑠 ∧ 𝑠 = {∅}) → 𝑥 ∈ {∅, 1𝑜})
56	50, 55	jca 554	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝑠 ∧ 𝑠 = {∅}) → (𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}))
57	56	orcd 407	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑠 ∧ 𝑠 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))
58	57	ex 450	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑠 → (𝑠 = {∅} → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
59		olc 399	. . . . . . . . . . . . . . . 16 ⊢ ((¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))
60	59	expcom 451	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑡 → (¬ 𝑡 = {∅} → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
61	58, 60	jaoa 532	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑠 ∨ 𝑥 ∈ 𝑡) → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
62	28, 61	sylbi 207	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑠 ∪ 𝑡) → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
63	62	adantl 482	. . . . . . . . . . . 12 ⊢ ((¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡)) → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
64	49, 63	jaoi 394	. . . . . . . . . . 11 ⊢ ((((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡))))
65		olc 399	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜})))
66	65	expcom 451	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {∅, 1𝑜} → (𝑡 = {∅} → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}))))
67	66	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) → (𝑡 = {∅} → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}))))
68	67	adantrd 484	. . . . . . . . . . . 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 450	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑠 = {∅} → (𝑥 ∈ 𝑠 → (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)))
71	70	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → (𝑥 ∈ 𝑠 → (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)))
72		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = {∅} → 𝑡 = {∅})
73	72, 52	syl6eqss 3655	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = {∅} → 𝑡 ⊆ {∅, 1𝑜})
74	73	sseld 3602	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = {∅} → (𝑥 ∈ 𝑡 → 𝑥 ∈ {∅, 1𝑜}))
75	74	anc2li 580	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = {∅} → (𝑥 ∈ 𝑡 → (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜})))
76	75	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → (𝑥 ∈ 𝑡 → (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜})))
77	71, 76	orim12d 883	. . . . . . . . . . . . . . 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 207	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑠 ∪ 𝑡) → ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}))))
80	79	adantl 482	. . . . . . . . . . . 12 ⊢ ((¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡)) → ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}))))
81	68, 80	jaoi 394	. . . . . . . . . . 11 ⊢ ((((𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ (𝑠 ∪ 𝑡) = {∅} ∧ 𝑥 ∈ (𝑠 ∪ 𝑡))) → ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → ((¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}))))
82	64, 81	orim12d 883	. . . . . . . . . 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 551	. . . . . . . . 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 241	. . . . . . . 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 997	. . . . . . . 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 396	. . . . . 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 232	. . . . 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 3753	. . . . . 6 ⊢ (𝑥 ∈ (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∪ if(𝑡 = {∅}, {∅, 1𝑜}, 𝑡)) ↔ (𝑥 ∈ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∨ 𝑥 ∈ if(𝑡 = {∅}, {∅, 1𝑜}, 𝑡)))
92		elif 4128	. . . . . . 7 ⊢ (𝑥 ∈ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ↔ ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥 ∈ 𝑠)))
93		elif 4128	. . . . . . 7 ⊢ (𝑥 ∈ if(𝑡 = {∅}, {∅, 1𝑜}, 𝑡) ↔ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1𝑜}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥 ∈ 𝑡)))
94	92, 93	orbi12i 543	. . . . . 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 226	. . . . 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 3609	. . 3 ⊢ ((𝑠 ∈ 𝒫 3𝑜 ∧ 𝑡 ∈ 𝒫 3𝑜) → if((𝑠 ∪ 𝑡) = {∅}, {∅, 1𝑜}, (𝑠 ∪ 𝑡)) ⊆ (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∪ if(𝑡 = {∅}, {∅, 1𝑜}, 𝑡)))
98		3on 7570	. . . . . 6 ⊢ 3𝑜 ∈ On
99	98	a1i 11	. . . . 5 ⊢ ((𝑠 ∈ 𝒫 3𝑜 ∧ 𝑡 ∈ 𝒫 3𝑜) → 3𝑜 ∈ On)
100		elpwi 4168	. . . . . 6 ⊢ (𝑠 ∈ 𝒫 3𝑜 → 𝑠 ⊆ 3𝑜)
101		elpwi 4168	. . . . . 6 ⊢ (𝑡 ∈ 𝒫 3𝑜 → 𝑡 ⊆ 3𝑜)
102		unss 3787	. . . . . . 7 ⊢ ((𝑠 ⊆ 3𝑜 ∧ 𝑡 ⊆ 3𝑜) ↔ (𝑠 ∪ 𝑡) ⊆ 3𝑜)
103	102	biimpi 206	. . . . . 6 ⊢ ((𝑠 ⊆ 3𝑜 ∧ 𝑡 ⊆ 3𝑜) → (𝑠 ∪ 𝑡) ⊆ 3𝑜)
104	100, 101, 103	syl2an 494	. . . . 5 ⊢ ((𝑠 ∈ 𝒫 3𝑜 ∧ 𝑡 ∈ 𝒫 3𝑜) → (𝑠 ∪ 𝑡) ⊆ 3𝑜)
105	99, 104	sselpwd 4807	. . . 4 ⊢ ((𝑠 ∈ 𝒫 3𝑜 ∧ 𝑡 ∈ 𝒫 3𝑜) → (𝑠 ∪ 𝑡) ∈ 𝒫 3𝑜)
106		eqeq1 2626	. . . . . 6 ⊢ (𝑟 = (𝑠 ∪ 𝑡) → (𝑟 = {∅} ↔ (𝑠 ∪ 𝑡) = {∅}))
107		id 22	. . . . . 6 ⊢ (𝑟 = (𝑠 ∪ 𝑡) → 𝑟 = (𝑠 ∪ 𝑡))
108	106, 107	ifbieq2d 4111	. . . . 5 ⊢ (𝑟 = (𝑠 ∪ 𝑡) → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if((𝑠 ∪ 𝑡) = {∅}, {∅, 1𝑜}, (𝑠 ∪ 𝑡)))
109		clsk1indlem.k	. . . . 5 ⊢ 𝐾 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))
110		prex 4909	. . . . . 6 ⊢ {∅, 1𝑜} ∈ V
111		vex 3203	. . . . . . 7 ⊢ 𝑠 ∈ V
112		vex 3203	. . . . . . 7 ⊢ 𝑡 ∈ V
113	111, 112	unex 6956	. . . . . 6 ⊢ (𝑠 ∪ 𝑡) ∈ V
114	110, 113	ifex 4156	. . . . 5 ⊢ if((𝑠 ∪ 𝑡) = {∅}, {∅, 1𝑜}, (𝑠 ∪ 𝑡)) ∈ V
115	108, 109, 114	fvmpt 6282	. . . 4 ⊢ ((𝑠 ∪ 𝑡) ∈ 𝒫 3𝑜 → (𝐾‘(𝑠 ∪ 𝑡)) = if((𝑠 ∪ 𝑡) = {∅}, {∅, 1𝑜}, (𝑠 ∪ 𝑡)))
116	105, 115	syl 17	. . 3 ⊢ ((𝑠 ∈ 𝒫 3𝑜 ∧ 𝑡 ∈ 𝒫 3𝑜) → (𝐾‘(𝑠 ∪ 𝑡)) = if((𝑠 ∪ 𝑡) = {∅}, {∅, 1𝑜}, (𝑠 ∪ 𝑡)))
117		eqeq1 2626	. . . . . . 7 ⊢ (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
118		id 22	. . . . . . 7 ⊢ (𝑟 = 𝑠 → 𝑟 = 𝑠)
119	117, 118	ifbieq2d 4111	. . . . . 6 ⊢ (𝑟 = 𝑠 → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
120	110, 111	ifex 4156	. . . . . 6 ⊢ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∈ V
121	119, 109, 120	fvmpt 6282	. . . . 5 ⊢ (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘𝑠) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
122	121	adantr 481	. . . 4 ⊢ ((𝑠 ∈ 𝒫 3𝑜 ∧ 𝑡 ∈ 𝒫 3𝑜) → (𝐾‘𝑠) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
123		eqeq1 2626	. . . . . . 7 ⊢ (𝑟 = 𝑡 → (𝑟 = {∅} ↔ 𝑡 = {∅}))
124		id 22	. . . . . . 7 ⊢ (𝑟 = 𝑡 → 𝑟 = 𝑡)
125	123, 124	ifbieq2d 4111	. . . . . 6 ⊢ (𝑟 = 𝑡 → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if(𝑡 = {∅}, {∅, 1𝑜}, 𝑡))
126	110, 112	ifex 4156	. . . . . 6 ⊢ if(𝑡 = {∅}, {∅, 1𝑜}, 𝑡) ∈ V
127	125, 109, 126	fvmpt 6282	. . . . 5 ⊢ (𝑡 ∈ 𝒫 3𝑜 → (𝐾‘𝑡) = if(𝑡 = {∅}, {∅, 1𝑜}, 𝑡))
128	127	adantl 482	. . . 4 ⊢ ((𝑠 ∈ 𝒫 3𝑜 ∧ 𝑡 ∈ 𝒫 3𝑜) → (𝐾‘𝑡) = if(𝑡 = {∅}, {∅, 1𝑜}, 𝑡))
129	122, 128	uneq12d 3768	. . 3 ⊢ ((𝑠 ∈ 𝒫 3𝑜 ∧ 𝑡 ∈ 𝒫 3𝑜) → ((𝐾‘𝑠) ∪ (𝐾‘𝑡)) = (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∪ if(𝑡 = {∅}, {∅, 1𝑜}, 𝑡)))
130	97, 116, 129	3sstr4d 3648	. 2 ⊢ ((𝑠 ∈ 𝒫 3𝑜 ∧ 𝑡 ∈ 𝒫 3𝑜) → (𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡)))
131	130	rgen2a 2977	1 ⊢ ∀𝑠 ∈ 𝒫 3𝑜∀𝑡 ∈ 𝒫 3𝑜(𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383   ∧ wa 384   ∨ w3o 1036   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  ifcif 4086  𝒫 cpw 4158  {csn 4177  {cpr 4179   ↦ cmpt 4729  Oncon0 5723  ‘cfv 5888  1_𝑜c1o 7553  3_𝑜c3o 7555

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-3or 1038  df-3an 1039  df-xor 1465  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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fv 5896  df-1o 7560  df-2o 7561  df-3o 7562

This theorem is referenced by:  clsk1independent  38344