Theorem pwuninel 8271

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 8270. (Contributed by NM, 27-Jun-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) Avoid ax-pr 5405 and ax-un 7733. (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 4908	. . 3 ⊢ (𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴)
2	1	sspwd 4580	. 2 ⊢ (𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ⊆ 𝒫 ∪ 𝐴)
3		pwnss 5323	. 2 ⊢ (𝒫 ∪ 𝐴 ∈ 𝐴 → ¬ 𝒫 𝒫 ∪ 𝐴 ⊆ 𝒫 ∪ 𝐴)
4	2, 3	pm2.65i 196	1 ⊢ ¬ 𝒫 ∪ 𝐴 ∈ 𝐴

Syntax hints:  ¬ wn 3   ∈ wcel 2149   ⊆ wss 3913  𝒫 cpw 4567  ∪ cuni 4876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-in 3920  df-ss 3930  df-pw 4569  df-uni 4877

This theorem is referenced by:  undefnel2  8274  disjen  9122  pnfnre  11250  kelac2lem  43717  kelac2  43718  ndfatafv2nrn  47881  afv2ndefb  47884