ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  0elpw GIF version

Theorem 0elpw 4166
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 3463 . 2 ∅ ⊆ 𝐴
2 0ex 4132 . . 3 ∅ ∈ V
32elpw 3583 . 2 (∅ ∈ 𝒫 𝐴 ↔ ∅ ⊆ 𝐴)
41, 3mpbir 146 1 ∅ ∈ 𝒫 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 2148  wss 3131  c0 3424  𝒫 cpw 3577
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-nul 4131
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-dif 3133  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579
This theorem is referenced by:  ordpwsucexmid  4571  pw1on  7227  pw1ne0  7229  pw1nct  14791
  Copyright terms: Public domain W3C validator