Theorem pwuninelOLD 8281

Description: Obsolete version of pwuninel 8280 as of 10-Jun-2026. (Contributed by NM, 27-Jun-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
pwuninelOLD	⊢ ¬ 𝒫 ∪ 𝐴 ∈ 𝐴

Proof of Theorem pwuninelOLD

Step	Hyp	Ref	Expression
1		pwexr 7773	. . 3 ⊢ (𝒫 ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 ∈ V)
2		pwuninel2 8279	. . 3 ⊢ (∪ 𝐴 ∈ V → ¬ 𝒫 ∪ 𝐴 ∈ 𝐴)
3	1, 2	syl 18	. 2 ⊢ (𝒫 ∪ 𝐴 ∈ 𝐴 → ¬ 𝒫 ∪ 𝐴 ∈ 𝐴)
4		id 23	. 2 ⊢ (¬ 𝒫 ∪ 𝐴 ∈ 𝐴 → ¬ 𝒫 ∪ 𝐴 ∈ 𝐴)
5	3, 4	pm2.61i 184	1 ⊢ ¬ 𝒫 ∪ 𝐴 ∈ 𝐴

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2146  Vcvv 3458  𝒫 cpw 4567  ∪ cuni 4877

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 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-un 3913  df-in 3915  df-ss 3925  df-pw 4569  df-sn 4595  df-pr 4597  df-uni 4878

This theorem is used by: (None)