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Theorem neipeltop 22026
Description: Lemma for neiptopreu 22030. (Contributed by Thierry Arnoux, 6-Jan-2018.)
Hypothesis
Ref Expression
neiptop.o 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
Assertion
Ref Expression
neipeltop (𝐶𝐽 ↔ (𝐶𝑋 ∧ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)))
Distinct variable groups:   𝑝,𝑎,𝐶   𝑁,𝑎   𝑋,𝑎
Allowed substitution hints:   𝐽(𝑝,𝑎)   𝑁(𝑝)   𝑋(𝑝)

Proof of Theorem neipeltop
StepHypRef Expression
1 eleq1 2825 . . . 4 (𝑎 = 𝐶 → (𝑎 ∈ (𝑁𝑝) ↔ 𝐶 ∈ (𝑁𝑝)))
21raleqbi1dv 3317 . . 3 (𝑎 = 𝐶 → (∀𝑝𝑎 𝑎 ∈ (𝑁𝑝) ↔ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)))
3 neiptop.o . . 3 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
42, 3elrab2 3605 . 2 (𝐶𝐽 ↔ (𝐶 ∈ 𝒫 𝑋 ∧ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)))
5 0ex 5200 . . . . . . 7 ∅ ∈ V
6 eleq1 2825 . . . . . . 7 (𝐶 = ∅ → (𝐶 ∈ V ↔ ∅ ∈ V))
75, 6mpbiri 261 . . . . . 6 (𝐶 = ∅ → 𝐶 ∈ V)
87adantl 485 . . . . 5 ((∀𝑝𝐶 𝐶 ∈ (𝑁𝑝) ∧ 𝐶 = ∅) → 𝐶 ∈ V)
9 elex 3426 . . . . . . 7 (𝐶 ∈ (𝑁𝑝) → 𝐶 ∈ V)
109ralimi 3083 . . . . . 6 (∀𝑝𝐶 𝐶 ∈ (𝑁𝑝) → ∀𝑝𝐶 𝐶 ∈ V)
11 r19.3rzv 4410 . . . . . . 7 (𝐶 ≠ ∅ → (𝐶 ∈ V ↔ ∀𝑝𝐶 𝐶 ∈ V))
1211biimparc 483 . . . . . 6 ((∀𝑝𝐶 𝐶 ∈ V ∧ 𝐶 ≠ ∅) → 𝐶 ∈ V)
1310, 12sylan 583 . . . . 5 ((∀𝑝𝐶 𝐶 ∈ (𝑁𝑝) ∧ 𝐶 ≠ ∅) → 𝐶 ∈ V)
148, 13pm2.61dane 3029 . . . 4 (∀𝑝𝐶 𝐶 ∈ (𝑁𝑝) → 𝐶 ∈ V)
15 elpwg 4516 . . . 4 (𝐶 ∈ V → (𝐶 ∈ 𝒫 𝑋𝐶𝑋))
1614, 15syl 17 . . 3 (∀𝑝𝐶 𝐶 ∈ (𝑁𝑝) → (𝐶 ∈ 𝒫 𝑋𝐶𝑋))
1716pm5.32ri 579 . 2 ((𝐶 ∈ 𝒫 𝑋 ∧ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)) ↔ (𝐶𝑋 ∧ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)))
184, 17bitri 278 1 (𝐶𝐽 ↔ (𝐶𝑋 ∧ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1543  wcel 2110  wne 2940  wral 3061  {crab 3065  Vcvv 3408  wss 3866  c0 4237  𝒫 cpw 4513  cfv 6380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2708  ax-nul 5199
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rab 3070  df-v 3410  df-dif 3869  df-in 3873  df-ss 3883  df-nul 4238  df-pw 4515
This theorem is referenced by:  neiptopuni  22027  neiptoptop  22028  neiptopnei  22029  neiptopreu  22030
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