Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clsk1indlem3 Structured version   Visualization version   GIF version

Theorem clsk1indlem3 40277
 Description: The ansatz closure function (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 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 3o𝑡 ∈ 𝒫 3o(𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))
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.
StepHypRef Expression
1 elif 4512 . . . . . 6 (𝑥 ∈ if((𝑠𝑡) = {∅}, {∅, 1o}, (𝑠𝑡)) ↔ (((𝑠𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠𝑡) = {∅} ∧ 𝑥 ∈ (𝑠𝑡))))
2 uneq12 4138 . . . . . . . . . . 11 ((𝑠 = {∅} ∧ 𝑡 = {∅}) → (𝑠𝑡) = ({∅} ∪ {∅}))
3 unidm 4132 . . . . . . . . . . 11 ({∅} ∪ {∅}) = {∅}
42, 3syl6eq 2877 . . . . . . . . . 10 ((𝑠 = {∅} ∧ 𝑡 = {∅}) → (𝑠𝑡) = {∅})
5 an3 655 . . . . . . . . . . . . . 14 (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∧ ((𝑠𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o})) → (𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}))
65orcd 871 . . . . . . . . . . . . 13 (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∧ ((𝑠𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o})) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)))
76orcd 871 . . . . . . . . . . . 12 (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∧ ((𝑠𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o})) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))
87ex 413 . . . . . . . . . . 11 ((𝑠 = {∅} ∧ 𝑡 = {∅}) → (((𝑠𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡)))))
9 pm2.24 124 . . . . . . . . . . . 12 ((𝑠𝑡) = {∅} → (¬ (𝑠𝑡) = {∅} → (𝑥 ∈ (𝑠𝑡) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))))
109impd 411 . . . . . . . . . . 11 ((𝑠𝑡) = {∅} → ((¬ (𝑠𝑡) = {∅} ∧ 𝑥 ∈ (𝑠𝑡)) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡)))))
118, 10jaao 950 . . . . . . . . . 10 (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∧ (𝑠𝑡) = {∅}) → ((((𝑠𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠𝑡) = {∅} ∧ 𝑥 ∈ (𝑠𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡)))))
124, 11mpdan 683 . . . . . . . . 9 ((𝑠 = {∅} ∧ 𝑡 = {∅}) → ((((𝑠𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠𝑡) = {∅} ∧ 𝑥 ∈ (𝑠𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡)))))
1312a1i 11 . . . . . . . 8 ((𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o) → ((𝑠 = {∅} ∧ 𝑡 = {∅}) → ((((𝑠𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠𝑡) = {∅} ∧ 𝑥 ∈ (𝑠𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))))
14 uneqsn 40257 . . . . . . . . . . . . 13 ((𝑠𝑡) = {∅} ↔ ((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅) ∨ (𝑠 = ∅ ∧ 𝑡 = {∅})))
15 df-3or 1082 . . . . . . . . . . . . 13 (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅) ∨ (𝑠 = ∅ ∧ 𝑡 = {∅})) ↔ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅)) ∨ (𝑠 = ∅ ∧ 𝑡 = {∅})))
1614, 15bitri 276 . . . . . . . . . . . 12 ((𝑠𝑡) = {∅} ↔ (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅)) ∨ (𝑠 = ∅ ∧ 𝑡 = {∅})))
17 pm2.21 123 . . . . . . . . . . . . . . . 16 𝑠 = {∅} → (𝑠 = {∅} → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))))
1817adantrd 492 . . . . . . . . . . . . . . 15 𝑠 = {∅} → ((𝑠 = {∅} ∧ 𝑡 = {∅}) → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))))
1917adantrd 492 . . . . . . . . . . . . . . 15 𝑠 = {∅} → ((𝑠 = {∅} ∧ 𝑡 = ∅) → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))))
2018, 19jaod 855 . . . . . . . . . . . . . 14 𝑠 = {∅} → (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅)) → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))))
2120adantr 481 . . . . . . . . . . . . 13 ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅)) → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))))
22 pm2.21 123 . . . . . . . . . . . . . . 15 𝑡 = {∅} → (𝑡 = {∅} → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))))
2322adantl 482 . . . . . . . . . . . . . 14 ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → (𝑡 = {∅} → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))))
2423adantld 491 . . . . . . . . . . . . 13 ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = ∅ ∧ 𝑡 = {∅}) → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))))
2521, 24jaod 855 . . . . . . . . . . . 12 ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (𝑠 = {∅} ∧ 𝑡 = ∅)) ∨ (𝑠 = ∅ ∧ 𝑡 = {∅})) → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))))
2616, 25syl5bi 243 . . . . . . . . . . 11 ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠𝑡) = {∅} → (𝑥 ∈ {∅, 1o} → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))))
2726impd 411 . . . . . . . . . 10 ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → (((𝑠𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡)))))
28 elun 4129 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝑠𝑡) ↔ (𝑥𝑠𝑥𝑡))
2928biimpi 217 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝑠𝑡) → (𝑥𝑠𝑥𝑡))
3029adantl 482 . . . . . . . . . . . . . 14 ((¬ (𝑠𝑡) = {∅} ∧ 𝑥 ∈ (𝑠𝑡)) → (𝑥𝑠𝑥𝑡))
31 andi 1003 . . . . . . . . . . . . . . 15 (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ (𝑥𝑠𝑥𝑡)) ↔ (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥𝑠) ∨ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥𝑡)))
32 simpl 483 . . . . . . . . . . . . . . . . 17 ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ¬ 𝑠 = {∅})
3332anim1i 614 . . . . . . . . . . . . . . . 16 (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥𝑠) → (¬ 𝑠 = {∅} ∧ 𝑥𝑠))
34 simpr 485 . . . . . . . . . . . . . . . . 17 ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ¬ 𝑡 = {∅})
3534anim1i 614 . . . . . . . . . . . . . . . 16 (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥𝑡) → (¬ 𝑡 = {∅} ∧ 𝑥𝑡))
3633, 35orim12i 904 . . . . . . . . . . . . . . 15 ((((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥𝑠) ∨ ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ 𝑥𝑡)) → ((¬ 𝑠 = {∅} ∧ 𝑥𝑠) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡)))
3731, 36sylbi 218 . . . . . . . . . . . . . 14 (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ (𝑥𝑠𝑥𝑡)) → ((¬ 𝑠 = {∅} ∧ 𝑥𝑠) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡)))
3830, 37sylan2 592 . . . . . . . . . . . . 13 (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ (¬ (𝑠𝑡) = {∅} ∧ 𝑥 ∈ (𝑠𝑡))) → ((¬ 𝑠 = {∅} ∧ 𝑥𝑠) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡)))
3938olcd 872 . . . . . . . . . . . 12 (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ (¬ (𝑠𝑡) = {∅} ∧ 𝑥 ∈ (𝑠𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o})) ∨ ((¬ 𝑠 = {∅} ∧ 𝑥𝑠) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))
40 or4 922 . . . . . . . . . . . 12 ((((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o})) ∨ ((¬ 𝑠 = {∅} ∧ 𝑥𝑠) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))) ↔ (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))
4139, 40sylib 219 . . . . . . . . . . 11 (((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∧ (¬ (𝑠𝑡) = {∅} ∧ 𝑥 ∈ (𝑠𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))
4241ex 413 . . . . . . . . . 10 ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((¬ (𝑠𝑡) = {∅} ∧ 𝑥 ∈ (𝑠𝑡)) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡)))))
4327, 42jaod 855 . . . . . . . . 9 ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((((𝑠𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠𝑡) = {∅} ∧ 𝑥 ∈ (𝑠𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡)))))
4443a1i 11 . . . . . . . 8 ((𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o) → ((¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((((𝑠𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠𝑡) = {∅} ∧ 𝑥 ∈ (𝑠𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))))
4513, 44jaod 855 . . . . . . 7 ((𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o) → (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅})) → ((((𝑠𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠𝑡) = {∅} ∧ 𝑥 ∈ (𝑠𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))))
46 orc 863 . . . . . . . . . . . . . . 15 ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡)))
4746expcom 414 . . . . . . . . . . . . . 14 (𝑥 ∈ {∅, 1o} → (𝑠 = {∅} → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))
4847adantrd 492 . . . . . . . . . . . . 13 (𝑥 ∈ {∅, 1o} → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))
4948adantl 482 . . . . . . . . . . . 12 (((𝑠𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))
50 simpr 485 . . . . . . . . . . . . . . . . . 18 ((𝑥𝑠𝑠 = {∅}) → 𝑠 = {∅})
51 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 = {∅} → 𝑠 = {∅})
52 snsspr1 4746 . . . . . . . . . . . . . . . . . . . . 21 {∅} ⊆ {∅, 1o}
5351, 52eqsstrdi 4025 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = {∅} → 𝑠 ⊆ {∅, 1o})
5453sseld 3970 . . . . . . . . . . . . . . . . . . 19 (𝑠 = {∅} → (𝑥𝑠𝑥 ∈ {∅, 1o}))
5554impcom 408 . . . . . . . . . . . . . . . . . 18 ((𝑥𝑠𝑠 = {∅}) → 𝑥 ∈ {∅, 1o})
5650, 55jca 512 . . . . . . . . . . . . . . . . 17 ((𝑥𝑠𝑠 = {∅}) → (𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}))
5756orcd 871 . . . . . . . . . . . . . . . 16 ((𝑥𝑠𝑠 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡)))
5857ex 413 . . . . . . . . . . . . . . 15 (𝑥𝑠 → (𝑠 = {∅} → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))
59 olc 864 . . . . . . . . . . . . . . . 16 ((¬ 𝑡 = {∅} ∧ 𝑥𝑡) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡)))
6059expcom 414 . . . . . . . . . . . . . . 15 (𝑥𝑡 → (¬ 𝑡 = {∅} → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))
6158, 60jaoa 951 . . . . . . . . . . . . . 14 ((𝑥𝑠𝑥𝑡) → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))
6228, 61sylbi 218 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑠𝑡) → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))
6362adantl 482 . . . . . . . . . . . 12 ((¬ (𝑠𝑡) = {∅} ∧ 𝑥 ∈ (𝑠𝑡)) → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))
6449, 63jaoi 853 . . . . . . . . . . 11 ((((𝑠𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠𝑡) = {∅} ∧ 𝑥 ∈ (𝑠𝑡))) → ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) → ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))
65 olc 864 . . . . . . . . . . . . . . 15 ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) → ((¬ 𝑠 = {∅} ∧ 𝑥𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o})))
6665expcom 414 . . . . . . . . . . . . . 14 (𝑥 ∈ {∅, 1o} → (𝑡 = {∅} → ((¬ 𝑠 = {∅} ∧ 𝑥𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
6766adantl 482 . . . . . . . . . . . . 13 (((𝑠𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) → (𝑡 = {∅} → ((¬ 𝑠 = {∅} ∧ 𝑥𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
6867adantrd 492 . . . . . . . . . . . 12 (((𝑠𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) → ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → ((¬ 𝑠 = {∅} ∧ 𝑥𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
69 id 22 . . . . . . . . . . . . . . . . . 18 ((¬ 𝑠 = {∅} ∧ 𝑥𝑠) → (¬ 𝑠 = {∅} ∧ 𝑥𝑠))
7069ex 413 . . . . . . . . . . . . . . . . 17 𝑠 = {∅} → (𝑥𝑠 → (¬ 𝑠 = {∅} ∧ 𝑥𝑠)))
7170adantl 482 . . . . . . . . . . . . . . . 16 ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → (𝑥𝑠 → (¬ 𝑠 = {∅} ∧ 𝑥𝑠)))
72 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = {∅} → 𝑡 = {∅})
7372, 52eqsstrdi 4025 . . . . . . . . . . . . . . . . . . 19 (𝑡 = {∅} → 𝑡 ⊆ {∅, 1o})
7473sseld 3970 . . . . . . . . . . . . . . . . . 18 (𝑡 = {∅} → (𝑥𝑡𝑥 ∈ {∅, 1o}))
7574anc2li 556 . . . . . . . . . . . . . . . . 17 (𝑡 = {∅} → (𝑥𝑡 → (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o})))
7675adantr 481 . . . . . . . . . . . . . . . 16 ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → (𝑥𝑡 → (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o})))
7771, 76orim12d 960 . . . . . . . . . . . . . . 15 ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → ((𝑥𝑠𝑥𝑡) → ((¬ 𝑠 = {∅} ∧ 𝑥𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
7877com12 32 . . . . . . . . . . . . . 14 ((𝑥𝑠𝑥𝑡) → ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → ((¬ 𝑠 = {∅} ∧ 𝑥𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
7928, 78sylbi 218 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑠𝑡) → ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → ((¬ 𝑠 = {∅} ∧ 𝑥𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
8079adantl 482 . . . . . . . . . . . 12 ((¬ (𝑠𝑡) = {∅} ∧ 𝑥 ∈ (𝑠𝑡)) → ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → ((¬ 𝑠 = {∅} ∧ 𝑥𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
8168, 80jaoi 853 . . . . . . . . . . 11 ((((𝑠𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠𝑡) = {∅} ∧ 𝑥 ∈ (𝑠𝑡))) → ((𝑡 = {∅} ∧ ¬ 𝑠 = {∅}) → ((¬ 𝑠 = {∅} ∧ 𝑥𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))))
8264, 81orim12d 960 . . . . . . . . . 10 ((((𝑠𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠𝑡) = {∅} ∧ 𝑥 ∈ (𝑠𝑡))) → (((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∨ (𝑡 = {∅} ∧ ¬ 𝑠 = {∅})) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡)) ∨ ((¬ 𝑠 = {∅} ∧ 𝑥𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o})))))
8382com12 32 . . . . . . . . 9 (((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∨ (𝑡 = {∅} ∧ ¬ 𝑠 = {∅})) → ((((𝑠𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠𝑡) = {∅} ∧ 𝑥 ∈ (𝑠𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡)) ∨ ((¬ 𝑠 = {∅} ∧ 𝑥𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o})))))
84 or42 923 . . . . . . . . 9 ((((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡)) ∨ ((¬ 𝑠 = {∅} ∧ 𝑥𝑠) ∨ (𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}))) ↔ (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))
8583, 84syl6ib 252 . . . . . . . 8 (((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∨ (𝑡 = {∅} ∧ ¬ 𝑠 = {∅})) → ((((𝑠𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠𝑡) = {∅} ∧ 𝑥 ∈ (𝑠𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡)))))
8685a1i 11 . . . . . . 7 ((𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o) → (((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∨ (𝑡 = {∅} ∧ ¬ 𝑠 = {∅})) → ((((𝑠𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠𝑡) = {∅} ∧ 𝑥 ∈ (𝑠𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))))
87 4exmid 1045 . . . . . . . 8 (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅})) ∨ ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∨ (𝑡 = {∅} ∧ ¬ 𝑠 = {∅})))
8887a1i 11 . . . . . . 7 ((𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o) → (((𝑠 = {∅} ∧ 𝑡 = {∅}) ∨ (¬ 𝑠 = {∅} ∧ ¬ 𝑡 = {∅})) ∨ ((𝑠 = {∅} ∧ ¬ 𝑡 = {∅}) ∨ (𝑡 = {∅} ∧ ¬ 𝑠 = {∅}))))
8945, 86, 88mpjaod 856 . . . . . 6 ((𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o) → ((((𝑠𝑡) = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ (𝑠𝑡) = {∅} ∧ 𝑥 ∈ (𝑠𝑡))) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡)))))
901, 89syl5bi 243 . . . . 5 ((𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o) → (𝑥 ∈ if((𝑠𝑡) = {∅}, {∅, 1o}, (𝑠𝑡)) → (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡)))))
91 elun 4129 . . . . . 6 (𝑥 ∈ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∪ if(𝑡 = {∅}, {∅, 1o}, 𝑡)) ↔ (𝑥 ∈ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∨ 𝑥 ∈ if(𝑡 = {∅}, {∅, 1o}, 𝑡)))
92 elif 4512 . . . . . . 7 (𝑥 ∈ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ ((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)))
93 elif 4512 . . . . . . 7 (𝑥 ∈ if(𝑡 = {∅}, {∅, 1o}, 𝑡) ↔ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡)))
9492, 93orbi12i 910 . . . . . 6 ((𝑥 ∈ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∨ 𝑥 ∈ if(𝑡 = {∅}, {∅, 1o}, 𝑡)) ↔ (((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))))
9591, 94sylbbr 237 . . . . 5 ((((𝑠 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑥𝑠)) ∨ ((𝑡 = {∅} ∧ 𝑥 ∈ {∅, 1o}) ∨ (¬ 𝑡 = {∅} ∧ 𝑥𝑡))) → 𝑥 ∈ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∪ if(𝑡 = {∅}, {∅, 1o}, 𝑡)))
9690, 95syl6 35 . . . 4 ((𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o) → (𝑥 ∈ if((𝑠𝑡) = {∅}, {∅, 1o}, (𝑠𝑡)) → 𝑥 ∈ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∪ if(𝑡 = {∅}, {∅, 1o}, 𝑡))))
9796ssrdv 3977 . . 3 ((𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o) → if((𝑠𝑡) = {∅}, {∅, 1o}, (𝑠𝑡)) ⊆ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∪ if(𝑡 = {∅}, {∅, 1o}, 𝑡)))
98 3on 8110 . . . . . 6 3o ∈ On
9998a1i 11 . . . . 5 ((𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o) → 3o ∈ On)
100 elpwi 4554 . . . . . 6 (𝑠 ∈ 𝒫 3o𝑠 ⊆ 3o)
101 elpwi 4554 . . . . . 6 (𝑡 ∈ 𝒫 3o𝑡 ⊆ 3o)
102 unss 4164 . . . . . . 7 ((𝑠 ⊆ 3o𝑡 ⊆ 3o) ↔ (𝑠𝑡) ⊆ 3o)
103102biimpi 217 . . . . . 6 ((𝑠 ⊆ 3o𝑡 ⊆ 3o) → (𝑠𝑡) ⊆ 3o)
104100, 101, 103syl2an 595 . . . . 5 ((𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o) → (𝑠𝑡) ⊆ 3o)
10599, 104sselpwd 5227 . . . 4 ((𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o) → (𝑠𝑡) ∈ 𝒫 3o)
106 eqeq1 2830 . . . . . 6 (𝑟 = (𝑠𝑡) → (𝑟 = {∅} ↔ (𝑠𝑡) = {∅}))
107 id 22 . . . . . 6 (𝑟 = (𝑠𝑡) → 𝑟 = (𝑠𝑡))
108106, 107ifbieq2d 4495 . . . . 5 (𝑟 = (𝑠𝑡) → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if((𝑠𝑡) = {∅}, {∅, 1o}, (𝑠𝑡)))
109 clsk1indlem.k . . . . 5 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))
110 prex 5329 . . . . . 6 {∅, 1o} ∈ V
111 vex 3503 . . . . . . 7 𝑠 ∈ V
112 vex 3503 . . . . . . 7 𝑡 ∈ V
113111, 112unex 7462 . . . . . 6 (𝑠𝑡) ∈ V
114110, 113ifex 4518 . . . . 5 if((𝑠𝑡) = {∅}, {∅, 1o}, (𝑠𝑡)) ∈ V
115108, 109, 114fvmpt 6767 . . . 4 ((𝑠𝑡) ∈ 𝒫 3o → (𝐾‘(𝑠𝑡)) = if((𝑠𝑡) = {∅}, {∅, 1o}, (𝑠𝑡)))
116105, 115syl 17 . . 3 ((𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o) → (𝐾‘(𝑠𝑡)) = if((𝑠𝑡) = {∅}, {∅, 1o}, (𝑠𝑡)))
117 eqeq1 2830 . . . . . . 7 (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
118 id 22 . . . . . . 7 (𝑟 = 𝑠𝑟 = 𝑠)
119117, 118ifbieq2d 4495 . . . . . 6 (𝑟 = 𝑠 → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
120110, 111ifex 4518 . . . . . 6 if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ V
121119, 109, 120fvmpt 6767 . . . . 5 (𝑠 ∈ 𝒫 3o → (𝐾𝑠) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
122121adantr 481 . . . 4 ((𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o) → (𝐾𝑠) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
123 eqeq1 2830 . . . . . . 7 (𝑟 = 𝑡 → (𝑟 = {∅} ↔ 𝑡 = {∅}))
124 id 22 . . . . . . 7 (𝑟 = 𝑡𝑟 = 𝑡)
125123, 124ifbieq2d 4495 . . . . . 6 (𝑟 = 𝑡 → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(𝑡 = {∅}, {∅, 1o}, 𝑡))
126110, 112ifex 4518 . . . . . 6 if(𝑡 = {∅}, {∅, 1o}, 𝑡) ∈ V
127125, 109, 126fvmpt 6767 . . . . 5 (𝑡 ∈ 𝒫 3o → (𝐾𝑡) = if(𝑡 = {∅}, {∅, 1o}, 𝑡))
128127adantl 482 . . . 4 ((𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o) → (𝐾𝑡) = if(𝑡 = {∅}, {∅, 1o}, 𝑡))
129122, 128uneq12d 4144 . . 3 ((𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o) → ((𝐾𝑠) ∪ (𝐾𝑡)) = (if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∪ if(𝑡 = {∅}, {∅, 1o}, 𝑡)))
13097, 116, 1293sstr4d 4018 . 2 ((𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o) → (𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡)))
131130rgen2a 3234 1 𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o(𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 843   ∨ w3o 1080   = wceq 1530   ∈ wcel 2107  ∀wral 3143   ∪ cun 3938   ⊆ wss 3940  ∅c0 4295  ifcif 4470  𝒫 cpw 4542  {csn 4564  {cpr 4566   ↦ cmpt 5143  Oncon0 6190  ‘cfv 6354  1oc1o 8091  3oc3o 8093 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326  ax-un 7455 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-xor 1498  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-ord 6193  df-on 6194  df-suc 6196  df-iota 6313  df-fun 6356  df-fv 6362  df-1o 8098  df-2o 8099  df-3o 8100 This theorem is referenced by:  clsk1independent  40280
 Copyright terms: Public domain W3C validator