Theorem pwne 4155

Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 3804. (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 4154	. 2 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴)
2		eqimss 3207	. . 3 ⊢ (𝒫 𝐴 = 𝐴 → 𝒫 𝐴 ⊆ 𝐴)
3	2	necon3bi 2395	. 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 2146   ≠ wne 2345   ⊆ wss 3127  𝒫 cpw 3572

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157  ax-sep 4116

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-rab 2462  df-v 2737  df-in 3133  df-ss 3140  df-pw 3574

This theorem is referenced by:  pnfnemnf  7986