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Theorem axpweq 4991
Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4992 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 4317 . . . 4 (𝒫 𝐴 ∈ V → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐴)
2 pweq 4305 . . . . . 6 (𝑥 = 𝒫 𝐴 → 𝒫 𝑥 = 𝒫 𝒫 𝐴)
32eleq2d 2825 . . . . 5 (𝑥 = 𝒫 𝐴 → (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ 𝒫 𝐴 ∈ 𝒫 𝒫 𝐴))
43spcegv 3434 . . . 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 3352 . . . 4 (𝒫 𝐴 ∈ 𝒫 𝑥 → 𝒫 𝐴 ∈ V)
76exlimiv 2007 . . 3 (∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥 → 𝒫 𝐴 ∈ V)
85, 7impbii 199 . 2 (𝒫 𝐴 ∈ V ↔ ∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥)
9 vex 3343 . . . . 5 𝑥 ∈ V
109elpw2 4977 . . . 4 (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ 𝒫 𝐴𝑥)
11 pwss 4319 . . . . 5 (𝒫 𝐴𝑥 ↔ ∀𝑦(𝑦𝐴𝑦𝑥))
12 dfss2 3732 . . . . . . 7 (𝑦𝐴 ↔ ∀𝑧(𝑧𝑦𝑧𝐴))
1312imbi1i 338 . . . . . 6 ((𝑦𝐴𝑦𝑥) ↔ (∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
1413albii 1896 . . . . 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 1923 . 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 1630   = wceq 1632  wex 1853  wcel 2139  Vcvv 3340  wss 3715  𝒫 cpw 4302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722  df-ss 3729  df-pw 4304
This theorem is referenced by: (None)
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