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Theorem 0elpw 3908
Description: Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
Assertion
Ref Expression
0elpw 𝒫 A

Proof of Theorem 0elpw
StepHypRef Expression
1 0ss 3249 . 2 ∅ ⊆ A
2 0ex 3875 . . 3 V
32elpw 3357 . 2 (∅ 𝒫 A ↔ ∅ ⊆ A)
41, 3mpbir 134 1 𝒫 A
Colors of variables: wff set class
Syntax hints:   wcel 1390  wss 2911  c0 3218  𝒫 cpw 3351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353
This theorem is referenced by:  ordpwsucexmid  4246
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