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Metamath Proof Explorer

Theorem pwuninel2

Description: Proof of pwuninel under the assumption that the union of the given class is a set, avoiding ax-pr and ax-un . (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 )