ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  axpweq GIF version

Theorem axpweq 4186
Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4189 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 3604 . . . 4 (𝒫 𝐴 ∈ V → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐴)
2 pweq 3593 . . . . . 6 (𝑥 = 𝒫 𝐴 → 𝒫 𝑥 = 𝒫 𝒫 𝐴)
32eleq2d 2259 . . . . 5 (𝑥 = 𝒫 𝐴 → (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ 𝒫 𝐴 ∈ 𝒫 𝒫 𝐴))
43spcegv 2840 . . . 4 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∈ 𝒫 𝒫 𝐴 → ∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥))
51, 4mpd 13 . . 3 (𝒫 𝐴 ∈ V → ∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥)
6 elex 2763 . . . 4 (𝒫 𝐴 ∈ 𝒫 𝑥 → 𝒫 𝐴 ∈ V)
76exlimiv 1609 . . 3 (∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥 → 𝒫 𝐴 ∈ V)
85, 7impbii 126 . 2 (𝒫 𝐴 ∈ V ↔ ∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥)
9 vex 2755 . . . . 5 𝑥 ∈ V
109elpw2 4172 . . . 4 (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ 𝒫 𝐴𝑥)
11 pwss 3606 . . . . 5 (𝒫 𝐴𝑥 ↔ ∀𝑦(𝑦𝐴𝑦𝑥))
12 dfss2 3159 . . . . . . 7 (𝑦𝐴 ↔ ∀𝑧(𝑧𝑦𝑧𝐴))
1312imbi1i 238 . . . . . 6 ((𝑦𝐴𝑦𝑥) ↔ (∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
1413albii 1481 . . . . 5 (∀𝑦(𝑦𝐴𝑦𝑥) ↔ ∀𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
1511, 14bitri 184 . . . 4 (𝒫 𝐴𝑥 ↔ ∀𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
1610, 15bitri 184 . . 3 (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ ∀𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
1716exbii 1616 . 2 (∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥 ↔ ∃𝑥𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
188, 17bitri 184 1 (𝒫 𝐴 ∈ V ↔ ∃𝑥𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1362   = wceq 1364  wex 1503  wcel 2160  Vcvv 2752  wss 3144  𝒫 cpw 3590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-sep 4136
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator