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Theorem grothpw 9851
 Description: Derive the Axiom of Power Sets ax-pow 4975 from the Tarski-Grothendieck axiom ax-groth 9848. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html. Note that ax-pow 4975 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 468 . . . . . . . 8 ((𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) → 𝒫 𝑧𝑦)
21ralimi 3101 . . . . . . 7 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) → ∀𝑧𝑦 𝒫 𝑧𝑦)
3 pweq 4301 . . . . . . . . 9 (𝑧 = 𝑥 → 𝒫 𝑧 = 𝒫 𝑥)
43sseq1d 3782 . . . . . . . 8 (𝑧 = 𝑥 → (𝒫 𝑧𝑦 ↔ 𝒫 𝑥𝑦))
54rspccv 3458 . . . . . . 7 (∀𝑧𝑦 𝒫 𝑧𝑦 → (𝑥𝑦 → 𝒫 𝑥𝑦))
62, 5syl 17 . . . . . 6 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) → (𝑥𝑦 → 𝒫 𝑥𝑦))
76anim2i 597 . . . . 5 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤)) → (𝑥𝑦 ∧ (𝑥𝑦 → 𝒫 𝑥𝑦)))
873adant3 1126 . . . 4 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) → (𝑥𝑦 ∧ (𝑥𝑦 → 𝒫 𝑥𝑦)))
9 pm3.35 798 . . . 4 ((𝑥𝑦 ∧ (𝑥𝑦 → 𝒫 𝑥𝑦)) → 𝒫 𝑥𝑦)
10 vex 3354 . . . . 5 𝑦 ∈ V
1110ssex 4937 . . . 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 9849 . . 3 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
1412, 13exlimiiv 2011 . 2 𝒫 𝑥 ∈ V
15 pwidg 4313 . . . . 5 (𝒫 𝑥 ∈ V → 𝒫 𝑥 ∈ 𝒫 𝒫 𝑥)
16 pweq 4301 . . . . . . 7 (𝑦 = 𝒫 𝑥 → 𝒫 𝑦 = 𝒫 𝒫 𝑥)
1716eleq2d 2836 . . . . . 6 (𝑦 = 𝒫 𝑥 → (𝒫 𝑥 ∈ 𝒫 𝑦 ↔ 𝒫 𝑥 ∈ 𝒫 𝒫 𝑥))
1817spcegv 3446 . . . . 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 3364 . . . . 5 (𝒫 𝑥 ∈ 𝒫 𝑦 → 𝒫 𝑥 ∈ V)
2120exlimiv 2010 . . . 4 (∃𝑦𝒫 𝑥 ∈ 𝒫 𝑦 → 𝒫 𝑥 ∈ V)
2219, 21impbii 199 . . 3 (𝒫 𝑥 ∈ V ↔ ∃𝑦𝒫 𝑥 ∈ 𝒫 𝑦)
2310elpw2 4960 . . . . 5 (𝒫 𝑥 ∈ 𝒫 𝑦 ↔ 𝒫 𝑥𝑦)
24 pwss 4315 . . . . . 6 (𝒫 𝑥𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))
25 dfss2 3741 . . . . . . . 8 (𝑧𝑥 ↔ ∀𝑤(𝑤𝑧𝑤𝑥))
2625imbi1i 338 . . . . . . 7 ((𝑧𝑥𝑧𝑦) ↔ (∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
2726albii 1895 . . . . . 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 1924 . . 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   ∧ wa 382   ∨ wo 828   ∧ w3a 1071  ∀wal 1629   = wceq 1631  ∃wex 1852   ∈ wcel 2145  ∀wral 3061  ∃wrex 3062  Vcvv 3351   ⊆ wss 3724  𝒫 cpw 4298   class class class wbr 4787   ≈ cen 8107 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-groth 9848 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-in 3731  df-ss 3738  df-pw 4300 This theorem is referenced by: (None)
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