Theorem clsk3nimkb 37820

Description: If the base set is not empty, axiom K3 does not imply KB. An concrete example with a pseudo-closure function of 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥)) is given. (Contributed by RP, 16-Jun-2021.)

Assertion

Ref	Expression
clsk3nimkb	⊢ ¬ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) → ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏))

Distinct variable group:   𝑘,𝑏,𝑡,𝑠

Proof of Theorem clsk3nimkb

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

Step	Hyp	Ref	Expression
1		1on 7512	. . . . . 6 ⊢ 1𝑜 ∈ On
2	1	elexi 3199	. . . . 5 ⊢ 1𝑜 ∈ V
3		1n0 7520	. . . . . 6 ⊢ 1𝑜 ≠ ∅
4		nelsn 4183	. . . . . 6 ⊢ (1𝑜 ≠ ∅ → ¬ 1𝑜 ∈ {∅})
5	3, 4	ax-mp 5	. . . . 5 ⊢ ¬ 1𝑜 ∈ {∅}
6		eldif 3565	. . . . . 6 ⊢ (1𝑜 ∈ (V ∖ {∅}) ↔ (1𝑜 ∈ V ∧ ¬ 1𝑜 ∈ {∅}))
7		ne0i 3897	. . . . . 6 ⊢ (1𝑜 ∈ (V ∖ {∅}) → (V ∖ {∅}) ≠ ∅)
8	6, 7	sylbir 225	. . . . 5 ⊢ ((1𝑜 ∈ V ∧ ¬ 1𝑜 ∈ {∅}) → (V ∖ {∅}) ≠ ∅)
9	2, 5, 8	mp2an 707	. . . 4 ⊢ (V ∖ {∅}) ≠ ∅
10		r19.2zb 4033	. . . 4 ⊢ ((V ∖ {∅}) ≠ ∅ ↔ (∀𝑏 ∈ (V ∖ {∅})∃𝑘 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ¬ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏)) → ∃𝑏 ∈ (V ∖ {∅})∃𝑘 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ¬ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏))))
11	9, 10	mpbi 220	. . 3 ⊢ (∀𝑏 ∈ (V ∖ {∅})∃𝑘 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ¬ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏)) → ∃𝑏 ∈ (V ∖ {∅})∃𝑘 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ¬ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏)))
12		rexex 2996	. . 3 ⊢ (∃𝑏 ∈ (V ∖ {∅})∃𝑘 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ¬ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏)) → ∃𝑏∃𝑘 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ¬ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏)))
13		rexanali 2992	. . . . 5 ⊢ (∃𝑘 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ¬ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏)) ↔ ¬ ∀𝑘 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) → ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏)))
14	13	exbii 1771	. . . 4 ⊢ (∃𝑏∃𝑘 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ¬ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏)) ↔ ∃𝑏 ¬ ∀𝑘 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) → ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏)))
15		exnal 1751	. . . 4 ⊢ (∃𝑏 ¬ ∀𝑘 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) → ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏)) ↔ ¬ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) → ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏)))
16	14, 15	sylbb 209	. . 3 ⊢ (∃𝑏∃𝑘 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ¬ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏)) → ¬ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) → ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏)))
17	11, 12, 16	3syl 18	. 2 ⊢ (∀𝑏 ∈ (V ∖ {∅})∃𝑘 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ¬ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏)) → ¬ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) → ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏)))
18		id 22	. . . . . . 7 ⊢ (𝑏 ∈ (V ∖ {∅}) → 𝑏 ∈ (V ∖ {∅}))
19		difssd 3716	. . . . . . 7 ⊢ (𝑏 ∈ (V ∖ {∅}) → (𝑏 ∖ 𝑥) ⊆ 𝑏)
20	18, 19	sselpwd 4767	. . . . . 6 ⊢ (𝑏 ∈ (V ∖ {∅}) → (𝑏 ∖ 𝑥) ∈ 𝒫 𝑏)
21	20	adantr 481	. . . . 5 ⊢ ((𝑏 ∈ (V ∖ {∅}) ∧ 𝑥 ∈ 𝒫 𝑏) → (𝑏 ∖ 𝑥) ∈ 𝒫 𝑏)
22		eqid 2621	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥)) = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))
23	21, 22	fmptd 6340	. . . 4 ⊢ (𝑏 ∈ (V ∖ {∅}) → (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥)):𝒫 𝑏⟶𝒫 𝑏)
24		pwexg 4810	. . . . 5 ⊢ (𝑏 ∈ (V ∖ {∅}) → 𝒫 𝑏 ∈ V)
25	24, 24	elmapd 7816	. . . 4 ⊢ (𝑏 ∈ (V ∖ {∅}) → ((𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥)) ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏) ↔ (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥)):𝒫 𝑏⟶𝒫 𝑏))
26	23, 25	mpbird 247	. . 3 ⊢ (𝑏 ∈ (V ∖ {∅}) → (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥)) ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏))
27		simpllr 798	. . . . . . . . 9 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥)))
28		difeq2 3700	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑏 ∖ 𝑥) = (𝑏 ∖ 𝑧))
29	28	cbvmptv 4710	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥)) = (𝑧 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑧))
30	27, 29	syl6eq 2671	. . . . . . . 8 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → 𝑘 = (𝑧 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑧)))
31		difeq2 3700	. . . . . . . . 9 ⊢ (𝑧 = (𝑠 ∪ 𝑡) → (𝑏 ∖ 𝑧) = (𝑏 ∖ (𝑠 ∪ 𝑡)))
32	31	adantl 482	. . . . . . . 8 ⊢ (((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) ∧ 𝑧 = (𝑠 ∪ 𝑡)) → (𝑏 ∖ 𝑧) = (𝑏 ∖ (𝑠 ∪ 𝑡)))
33		simplll 797	. . . . . . . . 9 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → 𝑏 ∈ (V ∖ {∅}))
34		simplr 791	. . . . . . . . . . 11 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → 𝑠 ∈ 𝒫 𝑏)
35	34	elpwid 4141	. . . . . . . . . 10 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → 𝑠 ⊆ 𝑏)
36		simpr 477	. . . . . . . . . . 11 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → 𝑡 ∈ 𝒫 𝑏)
37	36	elpwid 4141	. . . . . . . . . 10 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → 𝑡 ⊆ 𝑏)
38	35, 37	unssd 3767	. . . . . . . . 9 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → (𝑠 ∪ 𝑡) ⊆ 𝑏)
39	33, 38	sselpwd 4767	. . . . . . . 8 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → (𝑠 ∪ 𝑡) ∈ 𝒫 𝑏)
40		vex 3189	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
41	40	difexi 4769	. . . . . . . . 9 ⊢ (𝑏 ∖ (𝑠 ∪ 𝑡)) ∈ V
42	41	a1i 11	. . . . . . . 8 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → (𝑏 ∖ (𝑠 ∪ 𝑡)) ∈ V)
43	30, 32, 39, 42	fvmptd 6245	. . . . . . 7 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → (𝑘‘(𝑠 ∪ 𝑡)) = (𝑏 ∖ (𝑠 ∪ 𝑡)))
44		difeq2 3700	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑠 → (𝑏 ∖ 𝑧) = (𝑏 ∖ 𝑠))
45	44	adantl 482	. . . . . . . . . 10 ⊢ (((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) ∧ 𝑧 = 𝑠) → (𝑏 ∖ 𝑧) = (𝑏 ∖ 𝑠))
46	40	difexi 4769	. . . . . . . . . . 11 ⊢ (𝑏 ∖ 𝑠) ∈ V
47	46	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → (𝑏 ∖ 𝑠) ∈ V)
48	30, 45, 34, 47	fvmptd 6245	. . . . . . . . 9 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → (𝑘‘𝑠) = (𝑏 ∖ 𝑠))
49		difeq2 3700	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑡 → (𝑏 ∖ 𝑧) = (𝑏 ∖ 𝑡))
50	49	adantl 482	. . . . . . . . . 10 ⊢ (((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) ∧ 𝑧 = 𝑡) → (𝑏 ∖ 𝑧) = (𝑏 ∖ 𝑡))
51	40	difexi 4769	. . . . . . . . . . 11 ⊢ (𝑏 ∖ 𝑡) ∈ V
52	51	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → (𝑏 ∖ 𝑡) ∈ V)
53	30, 50, 36, 52	fvmptd 6245	. . . . . . . . 9 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → (𝑘‘𝑡) = (𝑏 ∖ 𝑡))
54	48, 53	uneq12d 3746	. . . . . . . 8 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = ((𝑏 ∖ 𝑠) ∪ (𝑏 ∖ 𝑡)))
55		difindi 3857	. . . . . . . 8 ⊢ (𝑏 ∖ (𝑠 ∩ 𝑡)) = ((𝑏 ∖ 𝑠) ∪ (𝑏 ∖ 𝑡))
56	54, 55	syl6eqr 2673	. . . . . . 7 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = (𝑏 ∖ (𝑠 ∩ 𝑡)))
57	43, 56	sseq12d 3613	. . . . . 6 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → ((𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ↔ (𝑏 ∖ (𝑠 ∪ 𝑡)) ⊆ (𝑏 ∖ (𝑠 ∩ 𝑡))))
58	57	ralbidva 2979	. . . . 5 ⊢ (((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) → (∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ↔ ∀𝑡 ∈ 𝒫 𝑏(𝑏 ∖ (𝑠 ∪ 𝑡)) ⊆ (𝑏 ∖ (𝑠 ∩ 𝑡))))
59	58	ralbidva 2979	. . . 4 ⊢ ((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) → (∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑏 ∖ (𝑠 ∪ 𝑡)) ⊆ (𝑏 ∖ (𝑠 ∩ 𝑡))))
60	56	eqeq1d 2623	. . . . . . . 8 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → (((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏 ↔ (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏))
61	60	imbi2d 330	. . . . . . 7 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → (((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏) ↔ ((𝑠 ∪ 𝑡) = 𝑏 → (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏)))
62	61	ralbidva 2979	. . . . . 6 ⊢ (((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) → (∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏) ↔ ∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏)))
63	62	ralbidva 2979	. . . . 5 ⊢ ((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) → (∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏) ↔ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏)))
64	63	notbid 308	. . . 4 ⊢ ((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) → (¬ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏) ↔ ¬ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏)))
65	59, 64	anbi12d 746	. . 3 ⊢ ((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) → ((∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ¬ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏)) ↔ (∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑏 ∖ (𝑠 ∪ 𝑡)) ⊆ (𝑏 ∖ (𝑠 ∩ 𝑡)) ∧ ¬ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏))))
66		pwidg 4144	. . . . . 6 ⊢ (𝑏 ∈ (V ∖ {∅}) → 𝑏 ∈ 𝒫 𝑏)
67		ssid 3603	. . . . . . 7 ⊢ 𝑏 ⊆ 𝑏
68	67	a1i 11	. . . . . 6 ⊢ (𝑏 ∈ (V ∖ {∅}) → 𝑏 ⊆ 𝑏)
69		eldifsni 4289	. . . . . . 7 ⊢ (𝑏 ∈ (V ∖ {∅}) → 𝑏 ≠ ∅)
70	69	neneqd 2795	. . . . . 6 ⊢ (𝑏 ∈ (V ∖ {∅}) → ¬ 𝑏 = ∅)
71		uneq1 3738	. . . . . . . . . 10 ⊢ (𝑠 = 𝑏 → (𝑠 ∪ 𝑡) = (𝑏 ∪ 𝑡))
72	71	eqeq1d 2623	. . . . . . . . 9 ⊢ (𝑠 = 𝑏 → ((𝑠 ∪ 𝑡) = 𝑏 ↔ (𝑏 ∪ 𝑡) = 𝑏))
73		ssequn2 3764	. . . . . . . . 9 ⊢ (𝑡 ⊆ 𝑏 ↔ (𝑏 ∪ 𝑡) = 𝑏)
74	72, 73	syl6bbr 278	. . . . . . . 8 ⊢ (𝑠 = 𝑏 → ((𝑠 ∪ 𝑡) = 𝑏 ↔ 𝑡 ⊆ 𝑏))
75		ineq1 3785	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑏 → (𝑠 ∩ 𝑡) = (𝑏 ∩ 𝑡))
76	75	difeq2d 3706	. . . . . . . . . 10 ⊢ (𝑠 = 𝑏 → (𝑏 ∖ (𝑠 ∩ 𝑡)) = (𝑏 ∖ (𝑏 ∩ 𝑡)))
77	76	eqeq1d 2623	. . . . . . . . 9 ⊢ (𝑠 = 𝑏 → ((𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏 ↔ (𝑏 ∖ (𝑏 ∩ 𝑡)) = 𝑏))
78	77	notbid 308	. . . . . . . 8 ⊢ (𝑠 = 𝑏 → (¬ (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏 ↔ ¬ (𝑏 ∖ (𝑏 ∩ 𝑡)) = 𝑏))
79	74, 78	anbi12d 746	. . . . . . 7 ⊢ (𝑠 = 𝑏 → (((𝑠 ∪ 𝑡) = 𝑏 ∧ ¬ (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏) ↔ (𝑡 ⊆ 𝑏 ∧ ¬ (𝑏 ∖ (𝑏 ∩ 𝑡)) = 𝑏)))
80		sseq1 3605	. . . . . . . 8 ⊢ (𝑡 = 𝑏 → (𝑡 ⊆ 𝑏 ↔ 𝑏 ⊆ 𝑏))
81		ineq2 3786	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑏 → (𝑏 ∩ 𝑡) = (𝑏 ∩ 𝑏))
82		inidm 3800	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∩ 𝑏) = 𝑏
83	81, 82	syl6eq 2671	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑏 → (𝑏 ∩ 𝑡) = 𝑏)
84	83	difeq2d 3706	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑏 → (𝑏 ∖ (𝑏 ∩ 𝑡)) = (𝑏 ∖ 𝑏))
85		difid 3922	. . . . . . . . . . . 12 ⊢ (𝑏 ∖ 𝑏) = ∅
86	84, 85	syl6eq 2671	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑏 → (𝑏 ∖ (𝑏 ∩ 𝑡)) = ∅)
87	86	eqeq1d 2623	. . . . . . . . . 10 ⊢ (𝑡 = 𝑏 → ((𝑏 ∖ (𝑏 ∩ 𝑡)) = 𝑏 ↔ ∅ = 𝑏))
88		eqcom 2628	. . . . . . . . . 10 ⊢ (∅ = 𝑏 ↔ 𝑏 = ∅)
89	87, 88	syl6bb 276	. . . . . . . . 9 ⊢ (𝑡 = 𝑏 → ((𝑏 ∖ (𝑏 ∩ 𝑡)) = 𝑏 ↔ 𝑏 = ∅))
90	89	notbid 308	. . . . . . . 8 ⊢ (𝑡 = 𝑏 → (¬ (𝑏 ∖ (𝑏 ∩ 𝑡)) = 𝑏 ↔ ¬ 𝑏 = ∅))
91	80, 90	anbi12d 746	. . . . . . 7 ⊢ (𝑡 = 𝑏 → ((𝑡 ⊆ 𝑏 ∧ ¬ (𝑏 ∖ (𝑏 ∩ 𝑡)) = 𝑏) ↔ (𝑏 ⊆ 𝑏 ∧ ¬ 𝑏 = ∅)))
92	79, 91	rspc2ev 3308	. . . . . 6 ⊢ ((𝑏 ∈ 𝒫 𝑏 ∧ 𝑏 ∈ 𝒫 𝑏 ∧ (𝑏 ⊆ 𝑏 ∧ ¬ 𝑏 = ∅)) → ∃𝑠 ∈ 𝒫 𝑏∃𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 ∧ ¬ (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏))
93	66, 66, 68, 70, 92	syl112anc 1327	. . . . 5 ⊢ (𝑏 ∈ (V ∖ {∅}) → ∃𝑠 ∈ 𝒫 𝑏∃𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 ∧ ¬ (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏))
94		rexanali 2992	. . . . . . 7 ⊢ (∃𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 ∧ ¬ (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏) ↔ ¬ ∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏))
95	94	rexbii 3034	. . . . . 6 ⊢ (∃𝑠 ∈ 𝒫 𝑏∃𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 ∧ ¬ (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏) ↔ ∃𝑠 ∈ 𝒫 𝑏 ¬ ∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏))
96		rexnal 2989	. . . . . 6 ⊢ (∃𝑠 ∈ 𝒫 𝑏 ¬ ∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏) ↔ ¬ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏))
97	95, 96	sylbb 209	. . . . 5 ⊢ (∃𝑠 ∈ 𝒫 𝑏∃𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 ∧ ¬ (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏) → ¬ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏))
98	93, 97	syl 17	. . . 4 ⊢ (𝑏 ∈ (V ∖ {∅}) → ¬ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏))
99		inss1 3811	. . . . . . 7 ⊢ (𝑠 ∩ 𝑡) ⊆ 𝑠
100		ssun1 3754	. . . . . . 7 ⊢ 𝑠 ⊆ (𝑠 ∪ 𝑡)
101	99, 100	sstri 3592	. . . . . 6 ⊢ (𝑠 ∩ 𝑡) ⊆ (𝑠 ∪ 𝑡)
102		sscon 3722	. . . . . 6 ⊢ ((𝑠 ∩ 𝑡) ⊆ (𝑠 ∪ 𝑡) → (𝑏 ∖ (𝑠 ∪ 𝑡)) ⊆ (𝑏 ∖ (𝑠 ∩ 𝑡)))
103	101, 102	ax-mp 5	. . . . 5 ⊢ (𝑏 ∖ (𝑠 ∪ 𝑡)) ⊆ (𝑏 ∖ (𝑠 ∩ 𝑡))
104	103	rgen2w 2920	. . . 4 ⊢ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑏 ∖ (𝑠 ∪ 𝑡)) ⊆ (𝑏 ∖ (𝑠 ∩ 𝑡))
105	98, 104	jctil 559	. . 3 ⊢ (𝑏 ∈ (V ∖ {∅}) → (∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑏 ∖ (𝑠 ∪ 𝑡)) ⊆ (𝑏 ∖ (𝑠 ∩ 𝑡)) ∧ ¬ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏)))
106	26, 65, 105	rspcedvd 3302	. 2 ⊢ (𝑏 ∈ (V ∖ {∅}) → ∃𝑘 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ¬ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏)))
107	17, 106	mprg 2921	1 ⊢ ¬ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) → ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384  ∀wal 1478   = wceq 1480  ∃wex 1701   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907  ∃wrex 2908  Vcvv 3186   ∖ cdif 3552   ∪ cun 3553   ∩ cin 3554   ⊆ wss 3555  ∅c0 3891  𝒫 cpw 4130  {csn 4148   ↦ cmpt 4673  Oncon0 5682  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604  1_𝑜c1o 7498   ↑_𝑚 cmap 7802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1o 7505  df-map 7804

This theorem is referenced by: (None)