Theorem pwuninel 8267

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 8266. (Contributed by NM, 27-Jun-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) Avoid ax-pr 5404 and ax-un 7732. (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 4904	. . 3 ⊢ (𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴)
2	1	sspwd 4575	. 2 ⊢ (𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ⊆ 𝒫 ∪ 𝐴)
3		pwnss 5322	. 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 2143   ⊆ wss 3905  𝒫 cpw 4562  ∪ cuni 4872

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1105  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-in 3912  df-ss 3922  df-pw 4564  df-uni 4873

This theorem is used by:  undefnel2  8270  disjen  9118  pnfnre  11254  kelac2lem  43819  kelac2  43820  ndfatafv2nrn  47986  afv2ndefb  47989