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Theorem pwuninel 7671
Description: The power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. See also pwuninel2 7670. (Contributed by NM, 27-Jun-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
Assertion
Ref Expression
pwuninel ¬ 𝒫 𝐴𝐴

Proof of Theorem pwuninel
StepHypRef Expression
1 pwexr 7239 . . 3 (𝒫 𝐴𝐴 𝐴 ∈ V)
2 pwuninel2 7670 . . 3 ( 𝐴 ∈ V → ¬ 𝒫 𝐴𝐴)
31, 2syl 17 . 2 (𝒫 𝐴𝐴 → ¬ 𝒫 𝐴𝐴)
4 id 22 . 2 (¬ 𝒫 𝐴𝐴 → ¬ 𝒫 𝐴𝐴)
53, 4pm2.61i 177 1 ¬ 𝒫 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2164  Vcvv 3414  𝒫 cpw 4380   cuni 4660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-nel 3103  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-pw 4382  df-sn 4400  df-pr 4402  df-uni 4661
This theorem is referenced by:  undefnel2  7673  disjen  8392  pnfnre  10405  kelac2lem  38472  kelac2  38473  ndfatafv2nrn  42117  afv2ndefb  42120
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