MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axpweq Structured version   Visualization version   GIF version

Theorem axpweq 4872
Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4873 is not used by the proof. (Contributed by NM, 22-Jun-2009.)
Hypothesis
Ref Expression
axpweq.1 𝐴 ∈ V
Assertion
Ref Expression
axpweq (𝒫 𝐴 ∈ V ↔ ∃𝑥𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem axpweq
StepHypRef Expression
1 pwidg 4206 . . . 4 (𝒫 𝐴 ∈ V → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐴)
2 pweq 4194 . . . . . 6 (𝑥 = 𝒫 𝐴 → 𝒫 𝑥 = 𝒫 𝒫 𝐴)
32eleq2d 2716 . . . . 5 (𝑥 = 𝒫 𝐴 → (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ 𝒫 𝐴 ∈ 𝒫 𝒫 𝐴))
43spcegv 3325 . . . 4 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∈ 𝒫 𝒫 𝐴 → ∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥))
51, 4mpd 15 . . 3 (𝒫 𝐴 ∈ V → ∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥)
6 elex 3243 . . . 4 (𝒫 𝐴 ∈ 𝒫 𝑥 → 𝒫 𝐴 ∈ V)
76exlimiv 1898 . . 3 (∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥 → 𝒫 𝐴 ∈ V)
85, 7impbii 199 . 2 (𝒫 𝐴 ∈ V ↔ ∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥)
9 vex 3234 . . . . 5 𝑥 ∈ V
109elpw2 4858 . . . 4 (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ 𝒫 𝐴𝑥)
11 pwss 4208 . . . . 5 (𝒫 𝐴𝑥 ↔ ∀𝑦(𝑦𝐴𝑦𝑥))
12 dfss2 3624 . . . . . . 7 (𝑦𝐴 ↔ ∀𝑧(𝑧𝑦𝑧𝐴))
1312imbi1i 338 . . . . . 6 ((𝑦𝐴𝑦𝑥) ↔ (∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
1413albii 1787 . . . . 5 (∀𝑦(𝑦𝐴𝑦𝑥) ↔ ∀𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
1511, 14bitri 264 . . . 4 (𝒫 𝐴𝑥 ↔ ∀𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
1610, 15bitri 264 . . 3 (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ ∀𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
1716exbii 1814 . 2 (∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥 ↔ ∃𝑥𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
188, 17bitri 264 1 (𝒫 𝐴 ∈ V ↔ ∃𝑥𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231  wss 3607  𝒫 cpw 4191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-in 3614  df-ss 3621  df-pw 4193
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator