Theorem pwne 4252

Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 3893. (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 4251	. 2 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴)
2		eqimss 3280	. . 3 ⊢ (𝒫 𝐴 = 𝐴 → 𝒫 𝐴 ⊆ 𝐴)
3	2	necon3bi 2451	. 2 ⊢ (¬ 𝒫 𝐴 ⊆ 𝐴 → 𝒫 𝐴 ≠ 𝐴)
4	1, 3	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≠ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2201   ≠ wne 2401   ⊆ wss 3199  𝒫 cpw 3653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212  ax-sep 4208

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-rab 2518  df-v 2803  df-in 3205  df-ss 3212  df-pw 3655

This theorem is referenced by:  pnfnemnf  8239