Theorem selpw 3389

Description: Setvar variable membership in a power class (common case). See elpw 3388. (Contributed by David A. Wheeler, 8-Dec-2018.)

Assertion

Ref	Expression
selpw	⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem selpw

Step	Hyp	Ref	Expression
1		vex 2604	. 2 ⊢ 𝑥 ∈ V
2	1	elpw 3388	1 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)

Syntax hints:   ↔ wb 103   ∈ wcel 1433   ⊆ wss 2973  𝒫 cpw 3382

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384

This theorem is referenced by:  ordpwsucss  4310  fabexg  5097  abexssex  5772  qsss  6188  npsspw  6661