Theorem pwuninel 8299

Description: The powerclass of the union of a class does not belong to that class. This theorem provides a way of constructing a new set that does not belong to a given set. See also pwuninel2 8298. (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 7784	. . 3 ⊢ (𝒫 ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 ∈ V)
2		pwuninel2 8298	. . 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 2106  Vcvv 3478  𝒫 cpw 4605  ∪ cuni 4912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-un 3968  df-in 3970  df-ss 3980  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913

This theorem is referenced by:  undefnel2  8301  disjen  9173  pnfnre  11300  kelac2lem  43053  kelac2  43054  ndfatafv2nrn  47171  afv2ndefb  47174