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Theorem neipeltop 21444
Description: Lemma for neiptopreu 21448. (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 2853 . . . 4 (𝑎 = 𝐶 → (𝑎 ∈ (𝑁𝑝) ↔ 𝐶 ∈ (𝑁𝑝)))
21raleqbi1dv 3343 . . 3 (𝑎 = 𝐶 → (∀𝑝𝑎 𝑎 ∈ (𝑁𝑝) ↔ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)))
3 neiptop.o . . 3 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
42, 3elrab2 3599 . 2 (𝐶𝐽 ↔ (𝐶 ∈ 𝒫 𝑋 ∧ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)))
5 0ex 5069 . . . . . . 7 ∅ ∈ V
6 eleq1 2853 . . . . . . 7 (𝐶 = ∅ → (𝐶 ∈ V ↔ ∅ ∈ V))
75, 6mpbiri 250 . . . . . 6 (𝐶 = ∅ → 𝐶 ∈ V)
87adantl 474 . . . . 5 ((∀𝑝𝐶 𝐶 ∈ (𝑁𝑝) ∧ 𝐶 = ∅) → 𝐶 ∈ V)
9 elex 3433 . . . . . . 7 (𝐶 ∈ (𝑁𝑝) → 𝐶 ∈ V)
109ralimi 3110 . . . . . 6 (∀𝑝𝐶 𝐶 ∈ (𝑁𝑝) → ∀𝑝𝐶 𝐶 ∈ V)
11 r19.3rzv 4328 . . . . . . 7 (𝐶 ≠ ∅ → (𝐶 ∈ V ↔ ∀𝑝𝐶 𝐶 ∈ V))
1211biimparc 472 . . . . . 6 ((∀𝑝𝐶 𝐶 ∈ V ∧ 𝐶 ≠ ∅) → 𝐶 ∈ V)
1310, 12sylan 572 . . . . 5 ((∀𝑝𝐶 𝐶 ∈ (𝑁𝑝) ∧ 𝐶 ≠ ∅) → 𝐶 ∈ V)
148, 13pm2.61dane 3055 . . . 4 (∀𝑝𝐶 𝐶 ∈ (𝑁𝑝) → 𝐶 ∈ V)
15 elpwg 4431 . . . 4 (𝐶 ∈ V → (𝐶 ∈ 𝒫 𝑋𝐶𝑋))
1614, 15syl 17 . . 3 (∀𝑝𝐶 𝐶 ∈ (𝑁𝑝) → (𝐶 ∈ 𝒫 𝑋𝐶𝑋))
1716pm5.32ri 568 . 2 ((𝐶 ∈ 𝒫 𝑋 ∧ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)) ↔ (𝐶𝑋 ∧ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)))
184, 17bitri 267 1 (𝐶𝐽 ↔ (𝐶𝑋 ∧ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387   = wceq 1507  wcel 2050  wne 2967  wral 3088  {crab 3092  Vcvv 3415  wss 3831  c0 4180  𝒫 cpw 4423  cfv 6190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750  ax-nul 5068
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rab 3097  df-v 3417  df-dif 3834  df-in 3838  df-ss 3845  df-nul 4181  df-pw 4425
This theorem is referenced by:  neiptopuni  21445  neiptoptop  21446  neiptopnei  21447  neiptopreu  21448
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