Theorem dfnrm2 49214

Description: A topological space is normal if any disjoint closed sets can be separated by open neighborhoods. An alternate definition of df-nrm 23263. (Contributed by Zhi Wang, 30-Aug-2024.)

Assertion

Ref	Expression
dfnrm2	⊢ Nrm = {𝑗 ∈ Top ∣ ∀𝑐 ∈ (Clsd‘𝑗)∀𝑑 ∈ (Clsd‘𝑗)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝑗 ∃𝑦 ∈ 𝑗 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))}

Distinct variable group:   𝑐,𝑑,𝑗,𝑥,𝑦

Proof of Theorem dfnrm2

Step	Hyp	Ref	Expression
1		isnrm3 23305	. . 3 ⊢ (𝑗 ∈ Nrm ↔ (𝑗 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝑗)∀𝑑 ∈ (Clsd‘𝑗)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝑗 ∃𝑦 ∈ 𝑗 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))))
2	1	eqabi 2870	. 2 ⊢ Nrm = {𝑗 ∣ (𝑗 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝑗)∀𝑑 ∈ (Clsd‘𝑗)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝑗 ∃𝑦 ∈ 𝑗 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))}
3		df-rab 3399	. 2 ⊢ {𝑗 ∈ Top ∣ ∀𝑐 ∈ (Clsd‘𝑗)∀𝑑 ∈ (Clsd‘𝑗)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝑗 ∃𝑦 ∈ 𝑗 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))} = {𝑗 ∣ (𝑗 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝑗)∀𝑑 ∈ (Clsd‘𝑗)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝑗 ∃𝑦 ∈ 𝑗 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))}
4	2, 3	eqtr4i 2761	1 ⊢ Nrm = {𝑗 ∈ Top ∣ ∀𝑐 ∈ (Clsd‘𝑗)∀𝑑 ∈ (Clsd‘𝑗)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝑗 ∃𝑦 ∈ 𝑗 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))}

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cab 2713  ∀wral 3050  ∃wrex 3059  {crab 3398   ∩ cin 3899   ⊆ wss 3900  ∅c0 4284  ‘cfv 6491  Topctop 22839  Clsdccld 22962  Nrmcnrm 23256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-iin 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-top 22840  df-cld 22965  df-cls 22967  df-nrm 23263

This theorem is referenced by: (None)