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Theorem grothpw 9504
Description: Derive the Axiom of Power Sets ax-pow 4764 from the Tarski-Grothendieck axiom ax-groth 9501. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html. Note that ax-pow 4764 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 471 . . . . . . . 8 ((𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) → 𝒫 𝑧𝑦)
21ralimi 2935 . . . . . . 7 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) → ∀𝑧𝑦 𝒫 𝑧𝑦)
3 pweq 4110 . . . . . . . . 9 (𝑧 = 𝑥 → 𝒫 𝑧 = 𝒫 𝑥)
43sseq1d 3594 . . . . . . . 8 (𝑧 = 𝑥 → (𝒫 𝑧𝑦 ↔ 𝒫 𝑥𝑦))
54rspccv 3278 . . . . . . 7 (∀𝑧𝑦 𝒫 𝑧𝑦 → (𝑥𝑦 → 𝒫 𝑥𝑦))
62, 5syl 17 . . . . . 6 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) → (𝑥𝑦 → 𝒫 𝑥𝑦))
76anim2i 590 . . . . 5 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤)) → (𝑥𝑦 ∧ (𝑥𝑦 → 𝒫 𝑥𝑦)))
873adant3 1073 . . . 4 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) → (𝑥𝑦 ∧ (𝑥𝑦 → 𝒫 𝑥𝑦)))
9 pm3.35 608 . . . 4 ((𝑥𝑦 ∧ (𝑥𝑦 → 𝒫 𝑥𝑦)) → 𝒫 𝑥𝑦)
10 vex 3175 . . . . 5 𝑦 ∈ V
1110ssex 4725 . . . 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 9502 . . 3 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
1412, 13exlimiiv 1845 . 2 𝒫 𝑥 ∈ V
15 pwidg 4120 . . . . 5 (𝒫 𝑥 ∈ V → 𝒫 𝑥 ∈ 𝒫 𝒫 𝑥)
16 pweq 4110 . . . . . . 7 (𝑦 = 𝒫 𝑥 → 𝒫 𝑦 = 𝒫 𝒫 𝑥)
1716eleq2d 2672 . . . . . 6 (𝑦 = 𝒫 𝑥 → (𝒫 𝑥 ∈ 𝒫 𝑦 ↔ 𝒫 𝑥 ∈ 𝒫 𝒫 𝑥))
1817spcegv 3266 . . . . 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 3184 . . . . 5 (𝒫 𝑥 ∈ 𝒫 𝑦 → 𝒫 𝑥 ∈ V)
2120exlimiv 1844 . . . 4 (∃𝑦𝒫 𝑥 ∈ 𝒫 𝑦 → 𝒫 𝑥 ∈ V)
2219, 21impbii 197 . . 3 (𝒫 𝑥 ∈ V ↔ ∃𝑦𝒫 𝑥 ∈ 𝒫 𝑦)
2310elpw2 4750 . . . . 5 (𝒫 𝑥 ∈ 𝒫 𝑦 ↔ 𝒫 𝑥𝑦)
24 pwss 4122 . . . . . 6 (𝒫 𝑥𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))
25 dfss2 3556 . . . . . . . 8 (𝑧𝑥 ↔ ∀𝑤(𝑤𝑧𝑤𝑥))
2625imbi1i 337 . . . . . . 7 ((𝑧𝑥𝑧𝑦) ↔ (∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
2726albii 1736 . . . . . 6 (∀𝑧(𝑧𝑥𝑧𝑦) ↔ ∀𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
2824, 27bitri 262 . . . . 5 (𝒫 𝑥𝑦 ↔ ∀𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
2923, 28bitri 262 . . . 4 (𝒫 𝑥 ∈ 𝒫 𝑦 ↔ ∀𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
3029exbii 1763 . . 3 (∃𝑦𝒫 𝑥 ∈ 𝒫 𝑦 ↔ ∃𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
3122, 30bitri 262 . 2 (𝒫 𝑥 ∈ V ↔ ∃𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
3214, 31mpbi 218 1 𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 381  wa 382  w3a 1030  wal 1472   = wceq 1474  wex 1694  wcel 1976  wral 2895  wrex 2896  Vcvv 3172  wss 3539  𝒫 cpw 4107   class class class wbr 4577  cen 7815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-groth 9501
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-v 3174  df-in 3546  df-ss 3553  df-pw 4109
This theorem is referenced by: (None)
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