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Theorem axpweq 5349
Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 5364 is not used by the proof. When ax-pow 5364 is assumed and 𝐴 is a set, both sides of the biconditional hold. In ZF, both sides hold if and only if 𝐴 is a set (see pwexr 7752). (Contributed by NM, 22-Jun-2009.)
Assertion
Ref Expression
axpweq (𝒫 𝐴 ∈ V ↔ ∃𝑥𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑧,𝐴,𝑦

Proof of Theorem axpweq
StepHypRef Expression
1 pwidg 4623 . . . 4 (𝒫 𝐴 ∈ V → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐴)
2 pweq 4617 . . . . . 6 (𝑥 = 𝒫 𝐴 → 𝒫 𝑥 = 𝒫 𝒫 𝐴)
32eleq2d 2820 . . . . 5 (𝑥 = 𝒫 𝐴 → (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ 𝒫 𝐴 ∈ 𝒫 𝒫 𝐴))
43spcegv 3588 . . . 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 3493 . . . 4 (𝒫 𝐴 ∈ 𝒫 𝑥 → 𝒫 𝐴 ∈ V)
76exlimiv 1934 . . 3 (∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥 → 𝒫 𝐴 ∈ V)
85, 7impbii 208 . 2 (𝒫 𝐴 ∈ V ↔ ∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥)
9 vex 3479 . . . . 5 𝑥 ∈ V
109elpw2 5346 . . . 4 (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ 𝒫 𝐴𝑥)
11 pwss 4626 . . . . 5 (𝒫 𝐴𝑥 ↔ ∀𝑦(𝑦𝐴𝑦𝑥))
12 dfss2 3969 . . . . . . 7 (𝑦𝐴 ↔ ∀𝑧(𝑧𝑦𝑧𝐴))
1312imbi1i 350 . . . . . 6 ((𝑦𝐴𝑦𝑥) ↔ (∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
1413albii 1822 . . . . 5 (∀𝑦(𝑦𝐴𝑦𝑥) ↔ ∀𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
1511, 14bitri 275 . . . 4 (𝒫 𝐴𝑥 ↔ ∀𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
1610, 15bitri 275 . . 3 (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ ∀𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
1716exbii 1851 . 2 (∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥 ↔ ∃𝑥𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
188, 17bitri 275 1 (𝒫 𝐴 ∈ V ↔ ∃𝑥𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1540   = wceq 1542  wex 1782  wcel 2107  Vcvv 3475  wss 3949  𝒫 cpw 4603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966  df-pw 4605
This theorem is referenced by:  grothpw  10821
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