Theorem ntrclsneine0 44071

Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that for every point, at least one (pseudo-)neighborbood exists hold equally. (Contributed by RP, 21-May-2021.)

Hypotheses

Ref	Expression
ntrcls.o	⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
ntrcls.d	⊢ 𝐷 = (𝑂‘𝐵)
ntrcls.r	⊢ (𝜑 → 𝐼𝐷𝐾)

Assertion

Ref	Expression
ntrclsneine0	⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∃𝑠 ∈ 𝒫 𝐵𝑥 ∈ (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑥 ∈ (𝐾‘𝑠)))

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑠   𝑗,𝐼,𝑘,𝑠   𝜑,𝑖,𝑗,𝑘,𝑠,𝑥

Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑥,𝑖,𝑗,𝑘,𝑠)   𝐼(𝑥,𝑖)   𝐾(𝑥,𝑖,𝑗,𝑘,𝑠)   𝑂(𝑥,𝑖,𝑗,𝑘,𝑠)

Proof of Theorem ntrclsneine0

Step	Hyp	Ref	Expression
1		ntrcls.o	. . 3 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
2		ntrcls.d	. . 3 ⊢ 𝐷 = (𝑂‘𝐵)
3		ntrcls.r	. . . 4 ⊢ (𝜑 → 𝐼𝐷𝐾)
4	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐼𝐷𝐾)
5		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
6	1, 2, 4, 5	ntrclsneine0lem 44070	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (∃𝑠 ∈ 𝒫 𝐵𝑥 ∈ (𝐼‘𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑥 ∈ (𝐾‘𝑠)))
7	6	ralbidva 3176	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∃𝑠 ∈ 𝒫 𝐵𝑥 ∈ (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑥 ∈ (𝐾‘𝑠)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  Vcvv 3481   ∖ cdif 3963  𝒫 cpw 4608   class class class wbr 5151   ↦ cmpt 5234  ‘cfv 6569  (class class class)co 7438   ↑_m cmap 8874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5288  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-iun 5001  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-f1 6574  df-fo 6575  df-f1o 6576  df-fv 6577  df-ov 7441  df-oprab 7442  df-mpo 7443  df-1st 8022  df-2nd 8023  df-map 8876

This theorem is referenced by: (None)