Metamath Proof Explorer


Theorem pwnss

Description: The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015)

Ref Expression
Assertion pwnss ( 𝐴𝑉 → ¬ 𝒫 𝐴𝐴 )

Proof

Step Hyp Ref Expression
1 rru ¬ { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ∈ 𝐴
2 ssel ( 𝒫 𝐴𝐴 → ( { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ∈ 𝒫 𝐴 → { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ∈ 𝐴 ) )
3 1 2 mtoi ( 𝒫 𝐴𝐴 → ¬ { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ∈ 𝒫 𝐴 )
4 ssrab2 { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ⊆ 𝐴
5 elpw2g ( 𝐴𝑉 → ( { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ∈ 𝒫 𝐴 ↔ { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ⊆ 𝐴 ) )
6 4 5 mpbiri ( 𝐴𝑉 → { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ∈ 𝒫 𝐴 )
7 3 6 nsyl3 ( 𝐴𝑉 → ¬ 𝒫 𝐴𝐴 )