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Theorem neipeltop 22854
Description: Lemma for neiptopreu 22858. (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 2820 . . . 4 (𝑎 = 𝐶 → (𝑎 ∈ (𝑁𝑝) ↔ 𝐶 ∈ (𝑁𝑝)))
21raleqbi1dv 3332 . . 3 (𝑎 = 𝐶 → (∀𝑝𝑎 𝑎 ∈ (𝑁𝑝) ↔ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)))
3 neiptop.o . . 3 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
42, 3elrab2 3687 . 2 (𝐶𝐽 ↔ (𝐶 ∈ 𝒫 𝑋 ∧ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)))
5 0ex 5308 . . . . . . 7 ∅ ∈ V
6 eleq1 2820 . . . . . . 7 (𝐶 = ∅ → (𝐶 ∈ V ↔ ∅ ∈ V))
75, 6mpbiri 257 . . . . . 6 (𝐶 = ∅ → 𝐶 ∈ V)
87adantl 481 . . . . 5 ((∀𝑝𝐶 𝐶 ∈ (𝑁𝑝) ∧ 𝐶 = ∅) → 𝐶 ∈ V)
9 elex 3492 . . . . . . 7 (𝐶 ∈ (𝑁𝑝) → 𝐶 ∈ V)
109ralimi 3082 . . . . . 6 (∀𝑝𝐶 𝐶 ∈ (𝑁𝑝) → ∀𝑝𝐶 𝐶 ∈ V)
11 r19.3rzv 4499 . . . . . . 7 (𝐶 ≠ ∅ → (𝐶 ∈ V ↔ ∀𝑝𝐶 𝐶 ∈ V))
1211biimparc 479 . . . . . 6 ((∀𝑝𝐶 𝐶 ∈ V ∧ 𝐶 ≠ ∅) → 𝐶 ∈ V)
1310, 12sylan 579 . . . . 5 ((∀𝑝𝐶 𝐶 ∈ (𝑁𝑝) ∧ 𝐶 ≠ ∅) → 𝐶 ∈ V)
148, 13pm2.61dane 3028 . . . 4 (∀𝑝𝐶 𝐶 ∈ (𝑁𝑝) → 𝐶 ∈ V)
15 elpwg 4606 . . . 4 (𝐶 ∈ V → (𝐶 ∈ 𝒫 𝑋𝐶𝑋))
1614, 15syl 17 . . 3 (∀𝑝𝐶 𝐶 ∈ (𝑁𝑝) → (𝐶 ∈ 𝒫 𝑋𝐶𝑋))
1716pm5.32ri 575 . 2 ((𝐶 ∈ 𝒫 𝑋 ∧ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)) ↔ (𝐶𝑋 ∧ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)))
184, 17bitri 274 1 (𝐶𝐽 ↔ (𝐶𝑋 ∧ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1540  wcel 2105  wne 2939  wral 3060  {crab 3431  Vcvv 3473  wss 3949  c0 4323  𝒫 cpw 4603  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2702  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3952  df-in 3956  df-ss 3966  df-nul 4324  df-pw 4605
This theorem is referenced by:  neiptopuni  22855  neiptoptop  22856  neiptopnei  22857  neiptopreu  22858
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