Theorem pwnss 5351

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

Assertion

Ref	Expression
pwnss	⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴)

Proof of Theorem pwnss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rru 3771	. . 3 ⊢ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴
2		ssel 3970	. . 3 ⊢ (𝒫 𝐴 ⊆ 𝐴 → ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝒫 𝐴 → {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴))
3	1, 2	mtoi 198	. 2 ⊢ (𝒫 𝐴 ⊆ 𝐴 → ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝒫 𝐴)
4		ssrab2 4073	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ⊆ 𝐴
5		elpw2g 5347	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝒫 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ⊆ 𝐴))
6	4, 5	mpbiri 257	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝒫 𝐴)
7	3, 6	nsyl3 138	1 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2098  {crab 3418   ⊆ wss 3944  𝒫 cpw 4604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300

This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-in 3951  df-ss 3961  df-pw 4606

This theorem is referenced by:  pwne  5352  pwuninel2  8280