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Theorem grothpw 10823
Description: Derive the Axiom of Power Sets ax-pow 5356 from the Tarski-Grothendieck axiom ax-groth 10820. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html 10820. Note that ax-pow 5356 is not used by the proof. (Contributed by GΓ©rard Lang, 22-Jun-2009.) (New usage is discouraged.)
Assertion
Ref Expression
grothpw βˆƒπ‘¦βˆ€π‘§(βˆ€π‘€(𝑀 ∈ 𝑧 β†’ 𝑀 ∈ π‘₯) β†’ 𝑧 ∈ 𝑦)
Distinct variable group:   π‘₯,𝑦,𝑧,𝑀

Proof of Theorem grothpw
StepHypRef Expression
1 simpl 482 . . . . . . . 8 ((𝒫 𝑧 βŠ† 𝑦 ∧ βˆƒπ‘€ ∈ 𝑦 𝒫 𝑧 βŠ† 𝑀) β†’ 𝒫 𝑧 βŠ† 𝑦)
21ralimi 3077 . . . . . . 7 (βˆ€π‘§ ∈ 𝑦 (𝒫 𝑧 βŠ† 𝑦 ∧ βˆƒπ‘€ ∈ 𝑦 𝒫 𝑧 βŠ† 𝑀) β†’ βˆ€π‘§ ∈ 𝑦 𝒫 𝑧 βŠ† 𝑦)
3 pweq 4611 . . . . . . . . 9 (𝑧 = π‘₯ β†’ 𝒫 𝑧 = 𝒫 π‘₯)
43sseq1d 4008 . . . . . . . 8 (𝑧 = π‘₯ β†’ (𝒫 𝑧 βŠ† 𝑦 ↔ 𝒫 π‘₯ βŠ† 𝑦))
54rspccv 3603 . . . . . . 7 (βˆ€π‘§ ∈ 𝑦 𝒫 𝑧 βŠ† 𝑦 β†’ (π‘₯ ∈ 𝑦 β†’ 𝒫 π‘₯ βŠ† 𝑦))
62, 5syl 17 . . . . . 6 (βˆ€π‘§ ∈ 𝑦 (𝒫 𝑧 βŠ† 𝑦 ∧ βˆƒπ‘€ ∈ 𝑦 𝒫 𝑧 βŠ† 𝑀) β†’ (π‘₯ ∈ 𝑦 β†’ 𝒫 π‘₯ βŠ† 𝑦))
76anim2i 616 . . . . 5 ((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 1129 . . . 4 ((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 pm3.35 800 . . . 4 ((π‘₯ ∈ 𝑦 ∧ (π‘₯ ∈ 𝑦 β†’ 𝒫 π‘₯ βŠ† 𝑦)) β†’ 𝒫 π‘₯ βŠ† 𝑦)
10 vex 3472 . . . . 5 𝑦 ∈ V
1110ssex 5314 . . . 4 (𝒫 π‘₯ βŠ† 𝑦 β†’ 𝒫 π‘₯ ∈ V)
128, 9, 113syl 18 . . 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†’ 𝒫 π‘₯ ∈ V)
13 axgroth5 10821 . . 3 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 1926 . 2 𝒫 π‘₯ ∈ V
15 axpweq 5341 . 2 (𝒫 π‘₯ ∈ V ↔ βˆƒπ‘¦βˆ€π‘§(βˆ€π‘€(𝑀 ∈ 𝑧 β†’ 𝑀 ∈ π‘₯) β†’ 𝑧 ∈ 𝑦))
1614, 15mpbi 229 1 βˆƒπ‘¦βˆ€π‘§(βˆ€π‘€(𝑀 ∈ 𝑧 β†’ 𝑀 ∈ π‘₯) β†’ 𝑧 ∈ 𝑦)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∨ wo 844   ∧ w3a 1084  βˆ€wal 1531  βˆƒwex 1773   ∈ wcel 2098  βˆ€wral 3055  βˆƒwrex 3064  Vcvv 3468   βŠ† wss 3943  π’« cpw 4597   class class class wbr 5141   β‰ˆ cen 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-groth 10820
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-in 3950  df-ss 3960  df-pw 4599
This theorem is referenced by: (None)
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