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Theorem grothpwex 9850
Description: Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 9846. Note that ax-pow 4971 is not used by the proof. Use axpweq 4970 to obtain ax-pow 4971. Use pwex 4976 or pwexg 4978 instead. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.)
Assertion
Ref Expression
grothpwex 𝒫 𝑥 ∈ V

Proof of Theorem grothpwex
Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 468 . . . . . . 7 ((𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) → 𝒫 𝑧𝑦)
21ralimi 3100 . . . . . 6 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) → ∀𝑧𝑦 𝒫 𝑧𝑦)
3 pweq 4298 . . . . . . . 8 (𝑧 = 𝑥 → 𝒫 𝑧 = 𝒫 𝑥)
43sseq1d 3779 . . . . . . 7 (𝑧 = 𝑥 → (𝒫 𝑧𝑦 ↔ 𝒫 𝑥𝑦))
54rspccv 3455 . . . . . 6 (∀𝑧𝑦 𝒫 𝑧𝑦 → (𝑥𝑦 → 𝒫 𝑥𝑦))
62, 5syl 17 . . . . 5 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) → (𝑥𝑦 → 𝒫 𝑥𝑦))
76anim2i 595 . . . 4 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤)) → (𝑥𝑦 ∧ (𝑥𝑦 → 𝒫 𝑥𝑦)))
873adant3 1125 . . 3 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) → (𝑥𝑦 ∧ (𝑥𝑦 → 𝒫 𝑥𝑦)))
9 pm3.35 810 . . 3 ((𝑥𝑦 ∧ (𝑥𝑦 → 𝒫 𝑥𝑦)) → 𝒫 𝑥𝑦)
10 vex 3352 . . . 4 𝑦 ∈ V
1110ssex 4933 . . 3 (𝒫 𝑥𝑦 → 𝒫 𝑥 ∈ V)
128, 9, 113syl 18 . 2 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) → 𝒫 𝑥 ∈ V)
13 axgroth5 9847 . 2 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
1412, 13exlimiiv 2010 1 𝒫 𝑥 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wo 826  w3a 1070  wcel 2144  wral 3060  wrex 3061  Vcvv 3349  wss 3721  𝒫 cpw 4295   class class class wbr 4784  cen 8105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-groth 9846
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-v 3351  df-in 3728  df-ss 3735  df-pw 4297
This theorem is referenced by:  isrnsigaOLD  30509
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