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Theorem grothpw 9608
Description: Derive the Axiom of Power Sets ax-pow 4813 from the Tarski-Grothendieck axiom ax-groth 9605. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html. Note that ax-pow 4813 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 473 . . . . . . . 8 ((𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) → 𝒫 𝑧𝑦)
21ralimi 2948 . . . . . . 7 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) → ∀𝑧𝑦 𝒫 𝑧𝑦)
3 pweq 4139 . . . . . . . . 9 (𝑧 = 𝑥 → 𝒫 𝑧 = 𝒫 𝑥)
43sseq1d 3617 . . . . . . . 8 (𝑧 = 𝑥 → (𝒫 𝑧𝑦 ↔ 𝒫 𝑥𝑦))
54rspccv 3296 . . . . . . 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 1079 . . . 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 3193 . . . . 5 𝑦 ∈ V
1110ssex 4772 . . . 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 9606 . . 3 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
1412, 13exlimiiv 1856 . 2 𝒫 𝑥 ∈ V
15 pwidg 4151 . . . . 5 (𝒫 𝑥 ∈ V → 𝒫 𝑥 ∈ 𝒫 𝒫 𝑥)
16 pweq 4139 . . . . . . 7 (𝑦 = 𝒫 𝑥 → 𝒫 𝑦 = 𝒫 𝒫 𝑥)
1716eleq2d 2684 . . . . . 6 (𝑦 = 𝒫 𝑥 → (𝒫 𝑥 ∈ 𝒫 𝑦 ↔ 𝒫 𝑥 ∈ 𝒫 𝒫 𝑥))
1817spcegv 3284 . . . . 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 3202 . . . . 5 (𝒫 𝑥 ∈ 𝒫 𝑦 → 𝒫 𝑥 ∈ V)
2120exlimiv 1855 . . . 4 (∃𝑦𝒫 𝑥 ∈ 𝒫 𝑦 → 𝒫 𝑥 ∈ V)
2219, 21impbii 199 . . 3 (𝒫 𝑥 ∈ V ↔ ∃𝑦𝒫 𝑥 ∈ 𝒫 𝑦)
2310elpw2 4798 . . . . 5 (𝒫 𝑥 ∈ 𝒫 𝑦 ↔ 𝒫 𝑥𝑦)
24 pwss 4153 . . . . . 6 (𝒫 𝑥𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))
25 dfss2 3577 . . . . . . . 8 (𝑧𝑥 ↔ ∀𝑤(𝑤𝑧𝑤𝑥))
2625imbi1i 339 . . . . . . 7 ((𝑧𝑥𝑧𝑦) ↔ (∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
2726albii 1744 . . . . . 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 1771 . . 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 383  wa 384  w3a 1036  wal 1478   = wceq 1480  wex 1701  wcel 1987  wral 2908  wrex 2909  Vcvv 3190  wss 3560  𝒫 cpw 4136   class class class wbr 4623  cen 7912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-groth 9605
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-v 3192  df-in 3567  df-ss 3574  df-pw 4138
This theorem is referenced by: (None)
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