Metamath Proof Explorer


Theorem neipeltop

Description: Lemma for neiptopreu . (Contributed by Thierry Arnoux, 6-Jan-2018)

Ref Expression
Hypothesis neiptop.o 𝐽 = { 𝑎 ∈ 𝒫 𝑋 ∣ ∀ 𝑝𝑎 𝑎 ∈ ( 𝑁𝑝 ) }
Assertion neipeltop ( 𝐶𝐽 ↔ ( 𝐶𝑋 ∧ ∀ 𝑝𝐶 𝐶 ∈ ( 𝑁𝑝 ) ) )

Proof

Step Hyp Ref Expression
1 neiptop.o 𝐽 = { 𝑎 ∈ 𝒫 𝑋 ∣ ∀ 𝑝𝑎 𝑎 ∈ ( 𝑁𝑝 ) }
2 eleq1 ( 𝑎 = 𝐶 → ( 𝑎 ∈ ( 𝑁𝑝 ) ↔ 𝐶 ∈ ( 𝑁𝑝 ) ) )
3 2 raleqbi1dv ( 𝑎 = 𝐶 → ( ∀ 𝑝𝑎 𝑎 ∈ ( 𝑁𝑝 ) ↔ ∀ 𝑝𝐶 𝐶 ∈ ( 𝑁𝑝 ) ) )
4 3 1 elrab2 ( 𝐶𝐽 ↔ ( 𝐶 ∈ 𝒫 𝑋 ∧ ∀ 𝑝𝐶 𝐶 ∈ ( 𝑁𝑝 ) ) )
5 0ex ∅ ∈ V
6 eleq1 ( 𝐶 = ∅ → ( 𝐶 ∈ V ↔ ∅ ∈ V ) )
7 5 6 mpbiri ( 𝐶 = ∅ → 𝐶 ∈ V )
8 7 adantl ( ( ∀ 𝑝𝐶 𝐶 ∈ ( 𝑁𝑝 ) ∧ 𝐶 = ∅ ) → 𝐶 ∈ V )
9 elex ( 𝐶 ∈ ( 𝑁𝑝 ) → 𝐶 ∈ V )
10 9 ralimi ( ∀ 𝑝𝐶 𝐶 ∈ ( 𝑁𝑝 ) → ∀ 𝑝𝐶 𝐶 ∈ V )
11 r19.3rzv ( 𝐶 ≠ ∅ → ( 𝐶 ∈ V ↔ ∀ 𝑝𝐶 𝐶 ∈ V ) )
12 11 biimparc ( ( ∀ 𝑝𝐶 𝐶 ∈ V ∧ 𝐶 ≠ ∅ ) → 𝐶 ∈ V )
13 10 12 sylan ( ( ∀ 𝑝𝐶 𝐶 ∈ ( 𝑁𝑝 ) ∧ 𝐶 ≠ ∅ ) → 𝐶 ∈ V )
14 8 13 pm2.61dane ( ∀ 𝑝𝐶 𝐶 ∈ ( 𝑁𝑝 ) → 𝐶 ∈ V )
15 elpwg ( 𝐶 ∈ V → ( 𝐶 ∈ 𝒫 𝑋𝐶𝑋 ) )
16 14 15 syl ( ∀ 𝑝𝐶 𝐶 ∈ ( 𝑁𝑝 ) → ( 𝐶 ∈ 𝒫 𝑋𝐶𝑋 ) )
17 16 pm5.32ri ( ( 𝐶 ∈ 𝒫 𝑋 ∧ ∀ 𝑝𝐶 𝐶 ∈ ( 𝑁𝑝 ) ) ↔ ( 𝐶𝑋 ∧ ∀ 𝑝𝐶 𝐶 ∈ ( 𝑁𝑝 ) ) )
18 4 17 bitri ( 𝐶𝐽 ↔ ( 𝐶𝑋 ∧ ∀ 𝑝𝐶 𝐶 ∈ ( 𝑁𝑝 ) ) )