Theorem pibt1 38339

Description: Theorem T000001 of pi-base. A compact topology is also countably compact. See pibp16 38336 and pibp19 38337 for the definitions of the relevant properties. (Contributed by ML, 8-Dec-2020.)

Hypothesis

Ref	Expression
pibt1.19	⊢ 𝐶 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥((∪ 𝑥 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)}

Assertion

Ref	Expression
pibt1	⊢ (𝐽 ∈ Comp → 𝐽 ∈ 𝐶)

Distinct variable group:   𝑥,𝐽,𝑦,𝑧

Allowed substitution hints:   𝐶(𝑥, 𝑦, 𝑧)

Proof of Theorem pibt1

Step	Hyp	Ref	Expression
1		pm3.41 498	. . . 4 ⊢ ((∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧) → ((∪ 𝐽 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧))
2	1	ralimi 3100	. . 3 ⊢ (∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧) → ∀𝑦 ∈ 𝒫 𝐽((∪ 𝐽 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧))
3	2	anim2i 629	. 2 ⊢ ((𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)) → (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽((∪ 𝐽 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)))
4		eqid 2761	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
5	4	pibp16 38336	. 2 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)))
6		pibt1.19	. . 3 ⊢ 𝐶 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥((∪ 𝑥 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)}
7	4, 6	pibp19 38337	. 2 ⊢ (𝐽 ∈ 𝐶 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽((∪ 𝐽 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)))
8	3, 5, 7	3imtr4i 295	1 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  {crab 3413   ∩ cin 3898  𝒫 cpw 4557  ∪ cuni 4867   class class class wbr 5103  ωcom 7877   ≼ cdom 8971  Fincfn 8973  Topctop 23211  Compccmp 23704

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-ss 3916  df-pw 4559  df-uni 4868  df-cmp 23705

This theorem is used by: (None)