Theorem neik0pk1imk0 41657

Description: Kuratowski's K0' and K1 axioms imply K0. Neighborhood version. (Contributed by RP, 3-Jun-2021.)

Hypotheses

Ref	Expression
neik0pk1imk0.bex	⊢ (𝜑 → 𝐵 ∈ 𝑉)
neik0pk1imk0.n	⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))
neik0pk1imk0.k0p	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ ∅)
neik0pk1imk0.k1	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥)))

Assertion

Ref	Expression
neik0pk1imk0	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥))

Distinct variable groups:   𝐵,𝑠,𝑡   𝑁,𝑠,𝑡   𝜑,𝑠,𝑥   𝑥,𝑡

Allowed substitution hints:   𝜑(𝑡)   𝐵(𝑥)   𝑁(𝑥)   𝑉(𝑥,𝑡,𝑠)

Proof of Theorem neik0pk1imk0

Step	Hyp	Ref	Expression
1		neik0pk1imk0.k1	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥)))
2		neik0pk1imk0.bex	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
3		pwidg 4555	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ 𝒫 𝐵)
4		sseq2 3947	. . . . . . . . . 10 ⊢ (𝑡 = 𝐵 → (𝑠 ⊆ 𝑡 ↔ 𝑠 ⊆ 𝐵))
5	4	anbi2d 629	. . . . . . . . 9 ⊢ (𝑡 = 𝐵 → ((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) ↔ (𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝐵)))
6		eleq1 2826	. . . . . . . . 9 ⊢ (𝑡 = 𝐵 → (𝑡 ∈ (𝑁‘𝑥) ↔ 𝐵 ∈ (𝑁‘𝑥)))
7	5, 6	imbi12d 345	. . . . . . . 8 ⊢ (𝑡 = 𝐵 → (((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥)) ↔ ((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝐵) → 𝐵 ∈ (𝑁‘𝑥))))
8	7	rspcv 3557	. . . . . . 7 ⊢ (𝐵 ∈ 𝒫 𝐵 → (∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥)) → ((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝐵) → 𝐵 ∈ (𝑁‘𝑥))))
9	2, 3, 8	3syl 18	. . . . . 6 ⊢ (𝜑 → (∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥)) → ((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝐵) → 𝐵 ∈ (𝑁‘𝑥))))
10	9	ralimdv 3109	. . . . 5 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥)) → ∀𝑠 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝐵) → 𝐵 ∈ (𝑁‘𝑥))))
11	10	ralimdv 3109	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥)) → ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝐵) → 𝐵 ∈ (𝑁‘𝑥))))
12	1, 11	mpd 15	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝐵) → 𝐵 ∈ (𝑁‘𝑥)))
13		r19.23v 3208	. . . . . 6 ⊢ (∀𝑠 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝐵) → 𝐵 ∈ (𝑁‘𝑥)) ↔ (∃𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝐵) → 𝐵 ∈ (𝑁‘𝑥)))
14	13	biimpi 215	. . . . 5 ⊢ (∀𝑠 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝐵) → 𝐵 ∈ (𝑁‘𝑥)) → (∃𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝐵) → 𝐵 ∈ (𝑁‘𝑥)))
15	14	a1i 11	. . . 4 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝐵) → 𝐵 ∈ (𝑁‘𝑥)) → (∃𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝐵) → 𝐵 ∈ (𝑁‘𝑥))))
16	15	ralimdv 3109	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝐵) → 𝐵 ∈ (𝑁‘𝑥)) → ∀𝑥 ∈ 𝐵 (∃𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝐵) → 𝐵 ∈ (𝑁‘𝑥))))
17	12, 16	mpd 15	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (∃𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝐵) → 𝐵 ∈ (𝑁‘𝑥)))
18		neik0pk1imk0.k0p	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ ∅)
19		neik0pk1imk0.n	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))
20		elmapi 8637	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
21	19, 20	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
22	21	ffvelrnda 6961	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑁‘𝑥) ∈ 𝒫 𝒫 𝐵)
23	22	elpwid 4544	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑁‘𝑥) ⊆ 𝒫 𝐵)
24	23	sseld 3920	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑠 ∈ (𝑁‘𝑥) → 𝑠 ∈ 𝒫 𝐵))
25	24	ancrd 552	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑠 ∈ (𝑁‘𝑥) → (𝑠 ∈ 𝒫 𝐵 ∧ 𝑠 ∈ (𝑁‘𝑥))))
26	25	eximdv 1920	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (∃𝑠 𝑠 ∈ (𝑁‘𝑥) → ∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ 𝑠 ∈ (𝑁‘𝑥))))
27		n0 4280	. . . . . . . 8 ⊢ ((𝑁‘𝑥) ≠ ∅ ↔ ∃𝑠 𝑠 ∈ (𝑁‘𝑥))
28		df-rex 3070	. . . . . . . 8 ⊢ (∃𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁‘𝑥) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ 𝑠 ∈ (𝑁‘𝑥)))
29	26, 27, 28	3imtr4g 296	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑁‘𝑥) ≠ ∅ → ∃𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁‘𝑥)))
30	29	imp 407	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑁‘𝑥) ≠ ∅) → ∃𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁‘𝑥))
31		elpwi 4542	. . . . . . . . 9 ⊢ (𝑠 ∈ 𝒫 𝐵 → 𝑠 ⊆ 𝐵)
32	24, 31	syl6 35	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑠 ∈ (𝑁‘𝑥) → 𝑠 ⊆ 𝐵))
33	32	ralrimivw 3104	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) → 𝑠 ⊆ 𝐵))
34	33	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑁‘𝑥) ≠ ∅) → ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) → 𝑠 ⊆ 𝐵))
35	30, 34	r19.29imd 3186	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑁‘𝑥) ≠ ∅) → ∃𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝐵))
36	35	ex 413	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑁‘𝑥) ≠ ∅ → ∃𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝐵)))
37	36	ralimdva 3108	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ ∅ → ∀𝑥 ∈ 𝐵 ∃𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝐵)))
38	18, 37	mpd 15	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∃𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝐵))
39		ralim 3083	. 2 ⊢ (∀𝑥 ∈ 𝐵 (∃𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝐵) → 𝐵 ∈ (𝑁‘𝑥)) → (∀𝑥 ∈ 𝐵 ∃𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝐵) → ∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥)))
40	17, 38, 39	sylc 65	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617

This theorem is referenced by: (None)