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Theorem grothpwex 10824
Description: Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 10820. Note that ax-pow 5362 is not used by the proof. Use axpweq 5347 to obtain ax-pow 5362. Use pwex 5377 or pwexg 5375 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 481 . . . . . . 7 ((𝒫 𝑧 βŠ† 𝑦 ∧ βˆƒπ‘€ ∈ 𝑦 𝒫 𝑧 βŠ† 𝑀) β†’ 𝒫 𝑧 βŠ† 𝑦)
21ralimi 3081 . . . . . 6 (βˆ€π‘§ ∈ 𝑦 (𝒫 𝑧 βŠ† 𝑦 ∧ βˆƒπ‘€ ∈ 𝑦 𝒫 𝑧 βŠ† 𝑀) β†’ βˆ€π‘§ ∈ 𝑦 𝒫 𝑧 βŠ† 𝑦)
3 pweq 4615 . . . . . . . 8 (𝑧 = π‘₯ β†’ 𝒫 𝑧 = 𝒫 π‘₯)
43sseq1d 4012 . . . . . . 7 (𝑧 = π‘₯ β†’ (𝒫 𝑧 βŠ† 𝑦 ↔ 𝒫 π‘₯ βŠ† 𝑦))
54rspccv 3608 . . . . . 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 615 . . . 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 1130 . . 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 pm3.35 799 . . 3 ((π‘₯ ∈ 𝑦 ∧ (π‘₯ ∈ 𝑦 β†’ 𝒫 π‘₯ βŠ† 𝑦)) β†’ 𝒫 π‘₯ βŠ† 𝑦)
10 vex 3476 . . . 4 𝑦 ∈ V
1110ssex 5320 . . 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 10821 . 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 1932 1 𝒫 π‘₯ ∈ V
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∨ wo 843   ∧ w3a 1085   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068  Vcvv 3472   βŠ† wss 3947  π’« cpw 4601   class class class wbr 5147   β‰ˆ cen 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-groth 10820
This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1087  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-in 3954  df-ss 3964  df-pw 4603
This theorem is referenced by: (None)
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