Theorem pwuninel 8062

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 8061. (Contributed by NM, 27-Jun-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)

Assertion

Ref	Expression
pwuninel	⊢ ¬ 𝒫 ∪ 𝐴 ∈ 𝐴

Proof of Theorem pwuninel

Step	Hyp	Ref	Expression
1		pwexr 7593	. . 3 ⊢ (𝒫 ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 ∈ V)
2		pwuninel2 8061	. . 3 ⊢ (∪ 𝐴 ∈ V → ¬ 𝒫 ∪ 𝐴 ∈ 𝐴)
3	1, 2	syl 17	. 2 ⊢ (𝒫 ∪ 𝐴 ∈ 𝐴 → ¬ 𝒫 ∪ 𝐴 ∈ 𝐴)
4		id 22	. 2 ⊢ (¬ 𝒫 ∪ 𝐴 ∈ 𝐴 → ¬ 𝒫 ∪ 𝐴 ∈ 𝐴)
5	3, 4	pm2.61i 182	1 ⊢ ¬ 𝒫 ∪ 𝐴 ∈ 𝐴

Syntax hints:  ¬ wn 3   ∈ wcel 2108  Vcvv 3422  𝒫 cpw 4530  ∪ cuni 4836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-pw 4532  df-sn 4559  df-pr 4561  df-uni 4837

This theorem is referenced by:  undefnel2  8064  disjen  8870  pnfnre  10947  kelac2lem  40805  kelac2  40806  ndfatafv2nrn  44600  afv2ndefb  44603