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Theorem grothpwex 9646
Description: Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 9642. Note that ax-pow 4841 is not used by the proof. Use axpweq 4840 to obtain ax-pow 4841. Use pwex 4846 or pwexg 4848 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 473 . . . . . . 7 ((𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) → 𝒫 𝑧𝑦)
21ralimi 2951 . . . . . 6 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) → ∀𝑧𝑦 𝒫 𝑧𝑦)
3 pweq 4159 . . . . . . . 8 (𝑧 = 𝑥 → 𝒫 𝑧 = 𝒫 𝑥)
43sseq1d 3630 . . . . . . 7 (𝑧 = 𝑥 → (𝒫 𝑧𝑦 ↔ 𝒫 𝑥𝑦))
54rspccv 3304 . . . . . 6 (∀𝑧𝑦 𝒫 𝑧𝑦 → (𝑥𝑦 → 𝒫 𝑥𝑦))
62, 5syl 17 . . . . 5 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) → (𝑥𝑦 → 𝒫 𝑥𝑦))
76anim2i 593 . . . 4 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤)) → (𝑥𝑦 ∧ (𝑥𝑦 → 𝒫 𝑥𝑦)))
873adant3 1080 . . 3 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) → (𝑥𝑦 ∧ (𝑥𝑦 → 𝒫 𝑥𝑦)))
9 pm3.35 611 . . 3 ((𝑥𝑦 ∧ (𝑥𝑦 → 𝒫 𝑥𝑦)) → 𝒫 𝑥𝑦)
10 vex 3201 . . . 4 𝑦 ∈ V
1110ssex 4800 . . 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 9643 . 2 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
1412, 13exlimiiv 1858 1 𝒫 𝑥 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384  w3a 1037  wcel 1989  wral 2911  wrex 2912  Vcvv 3198  wss 3572  𝒫 cpw 4156   class class class wbr 4651  cen 7949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-groth 9642
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-v 3200  df-in 3579  df-ss 3586  df-pw 4158
This theorem is referenced by:  isrnsigaOLD  30160
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