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Theorem axpweq 5000
Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 5001 is not used by the proof. When ax-pow 5001 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 7172). (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 4330 . . . 4 (𝒫 𝐴 ∈ V → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐴)
2 pweq 4318 . . . . . 6 (𝑥 = 𝒫 𝐴 → 𝒫 𝑥 = 𝒫 𝒫 𝐴)
32eleq2d 2830 . . . . 5 (𝑥 = 𝒫 𝐴 → (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ 𝒫 𝐴 ∈ 𝒫 𝒫 𝐴))
43spcegv 3446 . . . 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 3365 . . . 4 (𝒫 𝐴 ∈ 𝒫 𝑥 → 𝒫 𝐴 ∈ V)
76exlimiv 2025 . . 3 (∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥 → 𝒫 𝐴 ∈ V)
85, 7impbii 200 . 2 (𝒫 𝐴 ∈ V ↔ ∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥)
9 vex 3353 . . . . 5 𝑥 ∈ V
109elpw2 4986 . . . 4 (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ 𝒫 𝐴𝑥)
11 pwss 4332 . . . . 5 (𝒫 𝐴𝑥 ↔ ∀𝑦(𝑦𝐴𝑦𝑥))
12 dfss2 3749 . . . . . . 7 (𝑦𝐴 ↔ ∀𝑧(𝑧𝑦𝑧𝐴))
1312imbi1i 340 . . . . . 6 ((𝑦𝐴𝑦𝑥) ↔ (∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
1413albii 1914 . . . . 5 (∀𝑦(𝑦𝐴𝑦𝑥) ↔ ∀𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
1511, 14bitri 266 . . . 4 (𝒫 𝐴𝑥 ↔ ∀𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
1610, 15bitri 266 . . 3 (𝒫 𝐴 ∈ 𝒫 𝑥 ↔ ∀𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
1716exbii 1943 . 2 (∃𝑥𝒫 𝐴 ∈ 𝒫 𝑥 ↔ ∃𝑥𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
188, 17bitri 266 1 (𝒫 𝐴 ∈ V ↔ ∃𝑥𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1650   = wceq 1652  wex 1874  wcel 2155  Vcvv 3350  wss 3732  𝒫 cpw 4315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-in 3739  df-ss 3746  df-pw 4317
This theorem is referenced by: (None)
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