Theorem pwne 5108

Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 4710. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)

Assertion

Ref	Expression
pwne	⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≠ 𝐴)

Proof of Theorem pwne

Step	Hyp	Ref	Expression
1		pwnss 5107	. 2 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴)
2		eqimss 3915	. . 3 ⊢ (𝒫 𝐴 = 𝐴 → 𝒫 𝐴 ⊆ 𝐴)
3	2	necon3bi 2993	. 2 ⊢ (¬ 𝒫 𝐴 ⊆ 𝐴 → 𝒫 𝐴 ≠ 𝐴)
4	1, 3	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≠ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2050   ≠ wne 2967   ⊆ wss 3831  𝒫 cpw 4423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750  ax-sep 5061

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-rab 3097  df-v 3417  df-in 3838  df-ss 3845  df-pw 4425

This theorem is referenced by:  pnfnemnf  10498