Theorem pwuninel 8278

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 8277. (Contributed by NM, 27-Jun-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) Avoid ax-pr 5398 and ax-un 7742. (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 4899	. . 3 ⊢ (𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴)
2	1	sspwd 4570	. 2 ⊢ (𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ⊆ 𝒫 ∪ 𝐴)
3		pwnss 5316	. 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 2145   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-in 3906  df-ss 3916  df-pw 4559  df-uni 4868

This theorem is used by:  undefnel2  8281  disjen  9139  pnfnre  11299  kelac2lem  43970  kelac2  43971  ndfatafv2nrn  48174  afv2ndefb  48177