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Theorem 0elpw 3944
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 3282 . 2 ∅ ⊆ 𝐴
2 0ex 3911 . . 3 ∅ ∈ V
32elpw 3392 . 2 (∅ ∈ 𝒫 𝐴 ↔ ∅ ⊆ 𝐴)
41, 3mpbir 138 1 ∅ ∈ 𝒫 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 1409  wss 2944  c0 3251  𝒫 cpw 3386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-nul 3910
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2947  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388
This theorem is referenced by:  ordpwsucexmid  4321
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