Theorem isnrm4 48919

Description: A topological space is normal iff any two disjoint closed sets are separated by neighborhoods. (Contributed by Zhi Wang, 1-Sep-2024.)

Assertion

Ref	Expression
isnrm4	⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ ((nei‘𝐽)‘𝑐)∃𝑦 ∈ ((nei‘𝐽)‘𝑑)(𝑥 ∩ 𝑦) = ∅)))

Distinct variable group:   𝐽,𝑐,𝑑,𝑥,𝑦

Proof of Theorem isnrm4

Step	Hyp	Ref	Expression
1		isnrm3 23246	. 2 ⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))))
2		id 22	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝐽 ∈ Top)
3	2	sepnsepo 48912	. . . . 5 ⊢ (𝐽 ∈ Top → (∃𝑥 ∈ ((nei‘𝐽)‘𝑐)∃𝑦 ∈ ((nei‘𝐽)‘𝑑)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))
4	3	imbi2d 340	. . . 4 ⊢ (𝐽 ∈ Top → (((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ ((nei‘𝐽)‘𝑐)∃𝑦 ∈ ((nei‘𝐽)‘𝑑)(𝑥 ∩ 𝑦) = ∅) ↔ ((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))))
5	4	2ralbidv 3201	. . 3 ⊢ (𝐽 ∈ Top → (∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ ((nei‘𝐽)‘𝑐)∃𝑦 ∈ ((nei‘𝐽)‘𝑑)(𝑥 ∩ 𝑦) = ∅) ↔ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))))
6	5	pm5.32i 574	. 2 ⊢ ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ ((nei‘𝐽)‘𝑐)∃𝑦 ∈ ((nei‘𝐽)‘𝑑)(𝑥 ∩ 𝑦) = ∅)) ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))))
7	1, 6	bitr4i 278	1 ⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ ((nei‘𝐽)‘𝑐)∃𝑦 ∈ ((nei‘𝐽)‘𝑑)(𝑥 ∩ 𝑦) = ∅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  ‘cfv 6511  Topctop 22780  Clsdccld 22903  neicnei 22984  Nrmcnrm 23197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-top 22781  df-cld 22906  df-cls 22908  df-nei 22985  df-nrm 23204

This theorem is referenced by:  dfnrm3  48921