Theorem clsk3nimkb 40397

Description: If the base set is not empty, axiom K3 does not imply KB. A 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		1oex 8112	. . . . 5 ⊢ 1o ∈ V
2		1n0 8121	. . . . . 6 ⊢ 1o ≠ ∅
3		nelsn 4607	. . . . . 6 ⊢ (1o ≠ ∅ → ¬ 1o ∈ {∅})
4	2, 3	ax-mp 5	. . . . 5 ⊢ ¬ 1o ∈ {∅}
5		eldif 3948	. . . . . 6 ⊢ (1o ∈ (V ∖ {∅}) ↔ (1o ∈ V ∧ ¬ 1o ∈ {∅}))
6		ne0i 4302	. . . . . 6 ⊢ (1o ∈ (V ∖ {∅}) → (V ∖ {∅}) ≠ ∅)
7	5, 6	sylbir 237	. . . . 5 ⊢ ((1o ∈ V ∧ ¬ 1o ∈ {∅}) → (V ∖ {∅}) ≠ ∅)
8	1, 4, 7	mp2an 690	. . . 4 ⊢ (V ∖ {∅}) ≠ ∅
9		r19.2zb 4443	. . . 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))))
10	8, 9	mpbi 232	. . 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)))
11		rexex 3242	. . 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)))
12		rexanali 3267	. . . . 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)))
13	12	exbii 1848	. . . 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)))
14		exnal 1827	. . . 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	13, 14	sylbb 221	. . 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)))
16	10, 11, 15	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)))
17		difelpw 5254	. . . . . 6 ⊢ (𝑏 ∈ (V ∖ {∅}) → (𝑏 ∖ 𝑥) ∈ 𝒫 𝑏)
18	17	adantr 483	. . . . 5 ⊢ ((𝑏 ∈ (V ∖ {∅}) ∧ 𝑥 ∈ 𝒫 𝑏) → (𝑏 ∖ 𝑥) ∈ 𝒫 𝑏)
19	18	fmpttd 6881	. . . 4 ⊢ (𝑏 ∈ (V ∖ {∅}) → (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥)):𝒫 𝑏⟶𝒫 𝑏)
20		pwexg 5281	. . . . 5 ⊢ (𝑏 ∈ (V ∖ {∅}) → 𝒫 𝑏 ∈ V)
21	20, 20	elmapd 8422	. . . 4 ⊢ (𝑏 ∈ (V ∖ {∅}) → ((𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥)) ∈ (𝒫 𝑏 ↑m 𝒫 𝑏) ↔ (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥)):𝒫 𝑏⟶𝒫 𝑏))
22	19, 21	mpbird 259	. . 3 ⊢ (𝑏 ∈ (V ∖ {∅}) → (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥)) ∈ (𝒫 𝑏 ↑m 𝒫 𝑏))
23		simpllr 774	. . . . . . . . 9 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥)))
24		difeq2 4095	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑏 ∖ 𝑥) = (𝑏 ∖ 𝑧))
25	24	cbvmptv 5171	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥)) = (𝑧 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑧))
26	23, 25	syl6eq 2874	. . . . . . . 8 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → 𝑘 = (𝑧 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑧)))
27		difeq2 4095	. . . . . . . . 9 ⊢ (𝑧 = (𝑠 ∪ 𝑡) → (𝑏 ∖ 𝑧) = (𝑏 ∖ (𝑠 ∪ 𝑡)))
28	27	adantl 484	. . . . . . . 8 ⊢ (((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) ∧ 𝑧 = (𝑠 ∪ 𝑡)) → (𝑏 ∖ 𝑧) = (𝑏 ∖ (𝑠 ∪ 𝑡)))
29		simplll 773	. . . . . . . . 9 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → 𝑏 ∈ (V ∖ {∅}))
30		simplr 767	. . . . . . . . . . 11 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → 𝑠 ∈ 𝒫 𝑏)
31	30	elpwid 4552	. . . . . . . . . 10 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → 𝑠 ⊆ 𝑏)
32		simpr 487	. . . . . . . . . . 11 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → 𝑡 ∈ 𝒫 𝑏)
33	32	elpwid 4552	. . . . . . . . . 10 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → 𝑡 ⊆ 𝑏)
34	31, 33	unssd 4164	. . . . . . . . 9 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → (𝑠 ∪ 𝑡) ⊆ 𝑏)
35	29, 34	sselpwd 5232	. . . . . . . 8 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → (𝑠 ∪ 𝑡) ∈ 𝒫 𝑏)
36		vex 3499	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
37	36	difexi 5234	. . . . . . . . 9 ⊢ (𝑏 ∖ (𝑠 ∪ 𝑡)) ∈ V
38	37	a1i 11	. . . . . . . 8 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → (𝑏 ∖ (𝑠 ∪ 𝑡)) ∈ V)
39	26, 28, 35, 38	fvmptd 6777	. . . . . . 7 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → (𝑘‘(𝑠 ∪ 𝑡)) = (𝑏 ∖ (𝑠 ∪ 𝑡)))
40		difeq2 4095	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑠 → (𝑏 ∖ 𝑧) = (𝑏 ∖ 𝑠))
41	40	adantl 484	. . . . . . . . . 10 ⊢ (((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) ∧ 𝑧 = 𝑠) → (𝑏 ∖ 𝑧) = (𝑏 ∖ 𝑠))
42	36	difexi 5234	. . . . . . . . . . 11 ⊢ (𝑏 ∖ 𝑠) ∈ V
43	42	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → (𝑏 ∖ 𝑠) ∈ V)
44	26, 41, 30, 43	fvmptd 6777	. . . . . . . . 9 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → (𝑘‘𝑠) = (𝑏 ∖ 𝑠))
45		difeq2 4095	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑡 → (𝑏 ∖ 𝑧) = (𝑏 ∖ 𝑡))
46	45	adantl 484	. . . . . . . . . 10 ⊢ (((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) ∧ 𝑧 = 𝑡) → (𝑏 ∖ 𝑧) = (𝑏 ∖ 𝑡))
47	36	difexi 5234	. . . . . . . . . . 11 ⊢ (𝑏 ∖ 𝑡) ∈ V
48	47	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → (𝑏 ∖ 𝑡) ∈ V)
49	26, 46, 32, 48	fvmptd 6777	. . . . . . . . 9 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → (𝑘‘𝑡) = (𝑏 ∖ 𝑡))
50	44, 49	uneq12d 4142	. . . . . . . 8 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = ((𝑏 ∖ 𝑠) ∪ (𝑏 ∖ 𝑡)))
51		difindi 4260	. . . . . . . 8 ⊢ (𝑏 ∖ (𝑠 ∩ 𝑡)) = ((𝑏 ∖ 𝑠) ∪ (𝑏 ∖ 𝑡))
52	50, 51	syl6eqr 2876	. . . . . . 7 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = (𝑏 ∖ (𝑠 ∩ 𝑡)))
53	39, 52	sseq12d 4002	. . . . . 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))))
54	53	ralbidva 3198	. . . . 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))))
55	54	ralbidva 3198	. . . 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))))
56	52	eqeq1d 2825	. . . . . . . 8 ⊢ ((((𝑏 ∈ (V ∖ {∅}) ∧ 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥))) ∧ 𝑠 ∈ 𝒫 𝑏) ∧ 𝑡 ∈ 𝒫 𝑏) → (((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏 ↔ (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏))
57	56	imbi2d 343	. . . . . . 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)))
58	57	ralbidva 3198	. . . . . 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)))
59	58	ralbidva 3198	. . . . 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)))
60	59	notbid 320	. . . 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)))
61	55, 60	anbi12d 632	. . 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))))
62		pwidg 4563	. . . . . 6 ⊢ (𝑏 ∈ (V ∖ {∅}) → 𝑏 ∈ 𝒫 𝑏)
63		ssidd 3992	. . . . . 6 ⊢ (𝑏 ∈ (V ∖ {∅}) → 𝑏 ⊆ 𝑏)
64		eldifsnneq 4725	. . . . . 6 ⊢ (𝑏 ∈ (V ∖ {∅}) → ¬ 𝑏 = ∅)
65		uneq1 4134	. . . . . . . . . 10 ⊢ (𝑠 = 𝑏 → (𝑠 ∪ 𝑡) = (𝑏 ∪ 𝑡))
66	65	eqeq1d 2825	. . . . . . . . 9 ⊢ (𝑠 = 𝑏 → ((𝑠 ∪ 𝑡) = 𝑏 ↔ (𝑏 ∪ 𝑡) = 𝑏))
67		ssequn2 4161	. . . . . . . . 9 ⊢ (𝑡 ⊆ 𝑏 ↔ (𝑏 ∪ 𝑡) = 𝑏)
68	66, 67	syl6bbr 291	. . . . . . . 8 ⊢ (𝑠 = 𝑏 → ((𝑠 ∪ 𝑡) = 𝑏 ↔ 𝑡 ⊆ 𝑏))
69		ineq1 4183	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑏 → (𝑠 ∩ 𝑡) = (𝑏 ∩ 𝑡))
70	69	difeq2d 4101	. . . . . . . . . 10 ⊢ (𝑠 = 𝑏 → (𝑏 ∖ (𝑠 ∩ 𝑡)) = (𝑏 ∖ (𝑏 ∩ 𝑡)))
71	70	eqeq1d 2825	. . . . . . . . 9 ⊢ (𝑠 = 𝑏 → ((𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏 ↔ (𝑏 ∖ (𝑏 ∩ 𝑡)) = 𝑏))
72	71	notbid 320	. . . . . . . 8 ⊢ (𝑠 = 𝑏 → (¬ (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏 ↔ ¬ (𝑏 ∖ (𝑏 ∩ 𝑡)) = 𝑏))
73	68, 72	anbi12d 632	. . . . . . 7 ⊢ (𝑠 = 𝑏 → (((𝑠 ∪ 𝑡) = 𝑏 ∧ ¬ (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏) ↔ (𝑡 ⊆ 𝑏 ∧ ¬ (𝑏 ∖ (𝑏 ∩ 𝑡)) = 𝑏)))
74		sseq1 3994	. . . . . . . 8 ⊢ (𝑡 = 𝑏 → (𝑡 ⊆ 𝑏 ↔ 𝑏 ⊆ 𝑏))
75		ineq2 4185	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑏 → (𝑏 ∩ 𝑡) = (𝑏 ∩ 𝑏))
76		inidm 4197	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∩ 𝑏) = 𝑏
77	75, 76	syl6eq 2874	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑏 → (𝑏 ∩ 𝑡) = 𝑏)
78	77	difeq2d 4101	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑏 → (𝑏 ∖ (𝑏 ∩ 𝑡)) = (𝑏 ∖ 𝑏))
79		difid 4332	. . . . . . . . . . . 12 ⊢ (𝑏 ∖ 𝑏) = ∅
80	78, 79	syl6eq 2874	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑏 → (𝑏 ∖ (𝑏 ∩ 𝑡)) = ∅)
81	80	eqeq1d 2825	. . . . . . . . . 10 ⊢ (𝑡 = 𝑏 → ((𝑏 ∖ (𝑏 ∩ 𝑡)) = 𝑏 ↔ ∅ = 𝑏))
82		eqcom 2830	. . . . . . . . . 10 ⊢ (∅ = 𝑏 ↔ 𝑏 = ∅)
83	81, 82	syl6bb 289	. . . . . . . . 9 ⊢ (𝑡 = 𝑏 → ((𝑏 ∖ (𝑏 ∩ 𝑡)) = 𝑏 ↔ 𝑏 = ∅))
84	83	notbid 320	. . . . . . . 8 ⊢ (𝑡 = 𝑏 → (¬ (𝑏 ∖ (𝑏 ∩ 𝑡)) = 𝑏 ↔ ¬ 𝑏 = ∅))
85	74, 84	anbi12d 632	. . . . . . 7 ⊢ (𝑡 = 𝑏 → ((𝑡 ⊆ 𝑏 ∧ ¬ (𝑏 ∖ (𝑏 ∩ 𝑡)) = 𝑏) ↔ (𝑏 ⊆ 𝑏 ∧ ¬ 𝑏 = ∅)))
86	73, 85	rspc2ev 3637	. . . . . 6 ⊢ ((𝑏 ∈ 𝒫 𝑏 ∧ 𝑏 ∈ 𝒫 𝑏 ∧ (𝑏 ⊆ 𝑏 ∧ ¬ 𝑏 = ∅)) → ∃𝑠 ∈ 𝒫 𝑏∃𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 ∧ ¬ (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏))
87	62, 62, 63, 64, 86	syl112anc 1370	. . . . 5 ⊢ (𝑏 ∈ (V ∖ {∅}) → ∃𝑠 ∈ 𝒫 𝑏∃𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 ∧ ¬ (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏))
88		rexanali 3267	. . . . . . 7 ⊢ (∃𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 ∧ ¬ (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏) ↔ ¬ ∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏))
89	88	rexbii 3249	. . . . . 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))
90		rexnal 3240	. . . . . 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))
91	89, 90	sylbb 221	. . . . 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))
92	87, 91	syl 17	. . . 4 ⊢ (𝑏 ∈ (V ∖ {∅}) → ¬ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → (𝑏 ∖ (𝑠 ∩ 𝑡)) = 𝑏))
93		inss1 4207	. . . . . . 7 ⊢ (𝑠 ∩ 𝑡) ⊆ 𝑠
94		ssun1 4150	. . . . . . 7 ⊢ 𝑠 ⊆ (𝑠 ∪ 𝑡)
95	93, 94	sstri 3978	. . . . . 6 ⊢ (𝑠 ∩ 𝑡) ⊆ (𝑠 ∪ 𝑡)
96		sscon 4117	. . . . . 6 ⊢ ((𝑠 ∩ 𝑡) ⊆ (𝑠 ∪ 𝑡) → (𝑏 ∖ (𝑠 ∪ 𝑡)) ⊆ (𝑏 ∖ (𝑠 ∩ 𝑡)))
97	95, 96	ax-mp 5	. . . . 5 ⊢ (𝑏 ∖ (𝑠 ∪ 𝑡)) ⊆ (𝑏 ∖ (𝑠 ∩ 𝑡))
98	97	rgen2w 3153	. . . 4 ⊢ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑏 ∖ (𝑠 ∪ 𝑡)) ⊆ (𝑏 ∖ (𝑠 ∩ 𝑡))
99	92, 98	jctil 522	. . 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)))
100	22, 61, 99	rspcedvd 3628	. 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)))
101	16, 100	mprg 3154	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 398  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140  ∃wrex 3141  Vcvv 3496   ∖ cdif 3935   ∪ cun 3936   ∩ cin 3937   ⊆ wss 3938  ∅c0 4293  𝒫 cpw 4541  {csn 4569   ↦ cmpt 5148  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  1_oc1o 8097   ↑_m cmap 8408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ord 6196  df-on 6197  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-1o 8104  df-map 8410

This theorem is referenced by: (None)