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Theorem grothpw 9686
Description: Derive the Axiom of Power Sets ax-pow 4873 from the Tarski-Grothendieck axiom ax-groth 9683. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html. Note that ax-pow 4873 is not used by the proof. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.)
Assertion
Ref Expression
grothpw 𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Proof of Theorem grothpw
StepHypRef Expression
1 simpl 472 . . . . . . . 8 ((𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) → 𝒫 𝑧𝑦)
21ralimi 2981 . . . . . . 7 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) → ∀𝑧𝑦 𝒫 𝑧𝑦)
3 pweq 4194 . . . . . . . . 9 (𝑧 = 𝑥 → 𝒫 𝑧 = 𝒫 𝑥)
43sseq1d 3665 . . . . . . . 8 (𝑧 = 𝑥 → (𝒫 𝑧𝑦 ↔ 𝒫 𝑥𝑦))
54rspccv 3337 . . . . . . 7 (∀𝑧𝑦 𝒫 𝑧𝑦 → (𝑥𝑦 → 𝒫 𝑥𝑦))
62, 5syl 17 . . . . . 6 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) → (𝑥𝑦 → 𝒫 𝑥𝑦))
76anim2i 592 . . . . 5 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤)) → (𝑥𝑦 ∧ (𝑥𝑦 → 𝒫 𝑥𝑦)))
873adant3 1101 . . . 4 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) → (𝑥𝑦 ∧ (𝑥𝑦 → 𝒫 𝑥𝑦)))
9 pm3.35 610 . . . 4 ((𝑥𝑦 ∧ (𝑥𝑦 → 𝒫 𝑥𝑦)) → 𝒫 𝑥𝑦)
10 vex 3234 . . . . 5 𝑦 ∈ V
1110ssex 4835 . . . 4 (𝒫 𝑥𝑦 → 𝒫 𝑥 ∈ V)
128, 9, 113syl 18 . . 3 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) → 𝒫 𝑥 ∈ V)
13 axgroth5 9684 . . 3 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
1412, 13exlimiiv 1899 . 2 𝒫 𝑥 ∈ V
15 pwidg 4206 . . . . 5 (𝒫 𝑥 ∈ V → 𝒫 𝑥 ∈ 𝒫 𝒫 𝑥)
16 pweq 4194 . . . . . . 7 (𝑦 = 𝒫 𝑥 → 𝒫 𝑦 = 𝒫 𝒫 𝑥)
1716eleq2d 2716 . . . . . 6 (𝑦 = 𝒫 𝑥 → (𝒫 𝑥 ∈ 𝒫 𝑦 ↔ 𝒫 𝑥 ∈ 𝒫 𝒫 𝑥))
1817spcegv 3325 . . . . 5 (𝒫 𝑥 ∈ V → (𝒫 𝑥 ∈ 𝒫 𝒫 𝑥 → ∃𝑦𝒫 𝑥 ∈ 𝒫 𝑦))
1915, 18mpd 15 . . . 4 (𝒫 𝑥 ∈ V → ∃𝑦𝒫 𝑥 ∈ 𝒫 𝑦)
20 elex 3243 . . . . 5 (𝒫 𝑥 ∈ 𝒫 𝑦 → 𝒫 𝑥 ∈ V)
2120exlimiv 1898 . . . 4 (∃𝑦𝒫 𝑥 ∈ 𝒫 𝑦 → 𝒫 𝑥 ∈ V)
2219, 21impbii 199 . . 3 (𝒫 𝑥 ∈ V ↔ ∃𝑦𝒫 𝑥 ∈ 𝒫 𝑦)
2310elpw2 4858 . . . . 5 (𝒫 𝑥 ∈ 𝒫 𝑦 ↔ 𝒫 𝑥𝑦)
24 pwss 4208 . . . . . 6 (𝒫 𝑥𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))
25 dfss2 3624 . . . . . . . 8 (𝑧𝑥 ↔ ∀𝑤(𝑤𝑧𝑤𝑥))
2625imbi1i 338 . . . . . . 7 ((𝑧𝑥𝑧𝑦) ↔ (∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
2726albii 1787 . . . . . 6 (∀𝑧(𝑧𝑥𝑧𝑦) ↔ ∀𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
2824, 27bitri 264 . . . . 5 (𝒫 𝑥𝑦 ↔ ∀𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
2923, 28bitri 264 . . . 4 (𝒫 𝑥 ∈ 𝒫 𝑦 ↔ ∀𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
3029exbii 1814 . . 3 (∃𝑦𝒫 𝑥 ∈ 𝒫 𝑦 ↔ ∃𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
3122, 30bitri 264 . 2 (𝒫 𝑥 ∈ V ↔ ∃𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
3214, 31mpbi 220 1 𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383  w3a 1054  wal 1521   = wceq 1523  wex 1744  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  wss 3607  𝒫 cpw 4191   class class class wbr 4685  cen 7994
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  ax-groth 9683
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  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-ral 2946  df-rex 2947  df-v 3233  df-in 3614  df-ss 3621  df-pw 4193
This theorem is referenced by: (None)
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