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Theorem 0elpw 3891
 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 3232 . 2 ∅ ⊆ A
2 0ex 3858 . . 3 V
32elpw 3340 . 2 (∅ 𝒫 A ↔ ∅ ⊆ A)
41, 3mpbir 134 1 𝒫 A
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1374   ⊆ wss 2894  ∅c0 3201  𝒫 cpw 3334 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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-nul 3857 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-dif 2897  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336 This theorem is referenced by:  ordpwsucexmid  4230
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