Theorem pwuninel 8317

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 8316. (Contributed by NM, 27-Jun-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) Avoid ax-pr 5395 and ax-un 7772. (Revised by Umit Teoman Dogan, 10-Jun-2026.)

Assertion

Ref	Expression
pwuninel	⊢ ¬ 𝒫 ∪ 𝐴 ∈ 𝐴

Proof of Theorem pwuninel

Step	Hyp	Ref	Expression
1		elssuni 4902	. . 3 ⊢ (𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴)
2	1	sspwd 4573	. 2 ⊢ (𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ⊆ 𝒫 ∪ 𝐴)
3		pwnss 5317	. 2 ⊢ (𝒫 ∪ 𝐴 ∈ 𝐴 → ¬ 𝒫 𝒫 ∪ 𝐴 ⊆ 𝒫 ∪ 𝐴)
4	2, 3	pm2.65i 196	1 ⊢ ¬ 𝒫 ∪ 𝐴 ∈ 𝐴

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2147   ⊆ wss 3900  𝒫 cpw 4560  ∪ cuni 4870

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-pw 4562  df-uni 4871

This theorem is used by:  undefnel2  8320  disjen  9178  pnfnre  11374  kelac2lem  44257  kelac2  44258  ndfatafv2nrn  48494  afv2ndefb  48497