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Theorem grothpwex 10787
Description: Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 10783. Note that ax-pow 5340 is not used by the proof. Use axpweq 5325 to obtain ax-pow 5340. Use pwex 5355 or pwexg 5353 instead. (Contributed by GΓ©rard Lang, 22-Jun-2009.) (New usage is discouraged.)
Assertion
Ref Expression
grothpwex 𝒫 π‘₯ ∈ V
Proof of Theorem grothpwex
Dummy variables 𝑦 𝑧 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 483 . . . . . . 7 ((𝒫 𝑧 βŠ† 𝑦 ∧ βˆƒπ‘€ ∈ 𝑦 𝒫 𝑧 βŠ† 𝑀) β†’ 𝒫 𝑧 βŠ† 𝑦)
21ralimi 3082 . . . . . 6 (βˆ€π‘§ ∈ 𝑦 (𝒫 𝑧 βŠ† 𝑦 ∧ βˆƒπ‘€ ∈ 𝑦 𝒫 𝑧 βŠ† 𝑀) β†’ βˆ€π‘§ ∈ 𝑦 𝒫 𝑧 βŠ† 𝑦)
3 pweq 4594 . . . . . . . 8 (𝑧 = π‘₯ β†’ 𝒫 𝑧 = 𝒫 π‘₯)
43sseq1d 3993 . . . . . . 7 (𝑧 = π‘₯ β†’ (𝒫 𝑧 βŠ† 𝑦 ↔ 𝒫 π‘₯ βŠ† 𝑦))
54rspccv 3592 . . . . . 6 (βˆ€π‘§ ∈ 𝑦 𝒫 𝑧 βŠ† 𝑦 β†’ (π‘₯ ∈ 𝑦 β†’ 𝒫 π‘₯ βŠ† 𝑦))
62, 5syl 17 . . . . 5 (βˆ€π‘§ ∈ 𝑦 (𝒫 𝑧 βŠ† 𝑦 ∧ βˆƒπ‘€ ∈ 𝑦 𝒫 𝑧 βŠ† 𝑀) β†’ (π‘₯ ∈ 𝑦 β†’ 𝒫 π‘₯ βŠ† 𝑦))
76anim2i 617 . . . 4 ((π‘₯ ∈ 𝑦 ∧ βˆ€π‘§ ∈ 𝑦 (𝒫 𝑧 βŠ† 𝑦 ∧ βˆƒπ‘€ ∈ 𝑦 𝒫 𝑧 βŠ† 𝑀)) β†’ (π‘₯ ∈ 𝑦 ∧ (π‘₯ ∈ 𝑦 β†’ 𝒫 π‘₯ βŠ† 𝑦)))
873adant3 1132 . . 3 ((π‘₯ ∈ 𝑦 ∧ βˆ€π‘§ ∈ 𝑦 (𝒫 𝑧 βŠ† 𝑦 ∧ βˆƒπ‘€ ∈ 𝑦 𝒫 𝑧 βŠ† 𝑀) ∧ βˆ€π‘§ ∈ 𝒫 𝑦(𝑧 β‰ˆ 𝑦 ∨ 𝑧 ∈ 𝑦)) β†’ (π‘₯ ∈ 𝑦 ∧ (π‘₯ ∈ 𝑦 β†’ 𝒫 π‘₯ βŠ† 𝑦)))
9 pm3.35 801 . . 3 ((π‘₯ ∈ 𝑦 ∧ (π‘₯ ∈ 𝑦 β†’ 𝒫 π‘₯ βŠ† 𝑦)) β†’ 𝒫 π‘₯ βŠ† 𝑦)
10 vex 3463 . . . 4 𝑦 ∈ V
1110ssex 5298 . . 3 (𝒫 π‘₯ βŠ† 𝑦 β†’ 𝒫 π‘₯ ∈ V)
128, 9, 113syl 18 . 2 ((π‘₯ ∈ 𝑦 ∧ βˆ€π‘§ ∈ 𝑦 (𝒫 𝑧 βŠ† 𝑦 ∧ βˆƒπ‘€ ∈ 𝑦 𝒫 𝑧 βŠ† 𝑀) ∧ βˆ€π‘§ ∈ 𝒫 𝑦(𝑧 β‰ˆ 𝑦 ∨ 𝑧 ∈ 𝑦)) β†’ 𝒫 π‘₯ ∈ V)
13 axgroth5 10784 . 2 βˆƒπ‘¦(π‘₯ ∈ 𝑦 ∧ βˆ€π‘§ ∈ 𝑦 (𝒫 𝑧 βŠ† 𝑦 ∧ βˆƒπ‘€ ∈ 𝑦 𝒫 𝑧 βŠ† 𝑀) ∧ βˆ€π‘§ ∈ 𝒫 𝑦(𝑧 β‰ˆ 𝑦 ∨ 𝑧 ∈ 𝑦))
1412, 13exlimiiv 1934 1 𝒫 π‘₯ ∈ V
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   ∈ wcel 2106  βˆ€wral 3060  βˆƒwrex 3069  Vcvv 3459   βŠ† wss 3928  π’« cpw 4580   class class class wbr 5125   β‰ˆ cen 8902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5276  ax-groth 10783
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3419  df-v 3461  df-in 3935  df-ss 3945  df-pw 4582
This theorem is referenced by: (None)
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