Theorem pwuninel2

Description: Direct proof of pwuninel avoiding functions and thus several ZF axioms. (Contributed by Stefan O'Rear, 22-Feb-2015)

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

Proof

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