Theorem pwuninel2 8275

Description: Proof of pwuninel 8276 under the assumption that the union of the given class is a set, avoiding ax-pr 5391 and ax-un 7740. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
pwuninel2	⊢ (∪ 𝐴 ∈ 𝑉 → ¬ 𝒫 ∪ 𝐴 ∈ 𝐴)

Proof of Theorem pwuninel2

Step	Hyp	Ref	Expression
1		pwnss 5313	. 2 ⊢ (∪ 𝐴 ∈ 𝑉 → ¬ 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴)
2		elssuni 4899	. 2 ⊢ (𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴)
3	1, 2	nsyl 141	1 ⊢ (∪ 𝐴 ∈ 𝑉 → ¬ 𝒫 ∪ 𝐴 ∈ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ 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 2733  ax-sep 5249

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 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-in 3906  df-ss 3916  df-pw 4559  df-uni 4868

This theorem is used by:  pwuninelOLD  8277