Description: The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | pwnss | ⊢ ( 𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rru | ⊢ ¬ { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ∈ 𝐴 | |
2 | ssel | ⊢ ( 𝒫 𝐴 ⊆ 𝐴 → ( { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ∈ 𝒫 𝐴 → { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ∈ 𝐴 ) ) | |
3 | 1 2 | mtoi | ⊢ ( 𝒫 𝐴 ⊆ 𝐴 → ¬ { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ∈ 𝒫 𝐴 ) |
4 | ssrab2 | ⊢ { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ⊆ 𝐴 | |
5 | elpw2g | ⊢ ( 𝐴 ∈ 𝑉 → ( { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ∈ 𝒫 𝐴 ↔ { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ⊆ 𝐴 ) ) | |
6 | 4 5 | mpbiri | ⊢ ( 𝐴 ∈ 𝑉 → { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ∈ 𝒫 𝐴 ) |
7 | 3 6 | nsyl3 | ⊢ ( 𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴 ) |