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Theorem grothpw 9983
Description: Derive the Axiom of Power Sets ax-pow 5077 from the Tarski-Grothendieck axiom ax-groth 9980. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html. Note that ax-pow 5077 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 476 . . . . . . . 8 ((𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) → 𝒫 𝑧𝑦)
21ralimi 3134 . . . . . . 7 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) → ∀𝑧𝑦 𝒫 𝑧𝑦)
3 pweq 4382 . . . . . . . . 9 (𝑧 = 𝑥 → 𝒫 𝑧 = 𝒫 𝑥)
43sseq1d 3851 . . . . . . . 8 (𝑧 = 𝑥 → (𝒫 𝑧𝑦 ↔ 𝒫 𝑥𝑦))
54rspccv 3508 . . . . . . 7 (∀𝑧𝑦 𝒫 𝑧𝑦 → (𝑥𝑦 → 𝒫 𝑥𝑦))
62, 5syl 17 . . . . . 6 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) → (𝑥𝑦 → 𝒫 𝑥𝑦))
76anim2i 610 . . . . 5 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤)) → (𝑥𝑦 ∧ (𝑥𝑦 → 𝒫 𝑥𝑦)))
873adant3 1123 . . . 4 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) → (𝑥𝑦 ∧ (𝑥𝑦 → 𝒫 𝑥𝑦)))
9 pm3.35 793 . . . 4 ((𝑥𝑦 ∧ (𝑥𝑦 → 𝒫 𝑥𝑦)) → 𝒫 𝑥𝑦)
10 vex 3401 . . . . 5 𝑦 ∈ V
1110ssex 5039 . . . 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 9981 . . 3 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
1412, 13exlimiiv 1974 . 2 𝒫 𝑥 ∈ V
15 axpweq 5076 . 2 (𝒫 𝑥 ∈ V ↔ ∃𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
1614, 15mpbi 222 1 𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wo 836  w3a 1071  wal 1599  wex 1823  wcel 2107  wral 3090  wrex 3091  Vcvv 3398  wss 3792  𝒫 cpw 4379   class class class wbr 4886  cen 8238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754  ax-sep 5017  ax-groth 9980
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-v 3400  df-in 3799  df-ss 3806  df-pw 4381
This theorem is referenced by: (None)
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