MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  snelpw Structured version   Visualization version   GIF version

Theorem snelpw 4876
Description: A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.)
Hypothesis
Ref Expression
snelpw.1 𝐴 ∈ V
Assertion
Ref Expression
snelpw (𝐴𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)

Proof of Theorem snelpw
StepHypRef Expression
1 snelpw.1 . . 3 𝐴 ∈ V
21snss 4288 . 2 (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵)
3 snex 4871 . . 3 {𝐴} ∈ V
43elpw 4138 . 2 ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵)
52, 4bitr4i 267 1 (𝐴𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 1987  Vcvv 3186  wss 3556  𝒫 cpw 4132  {csn 4150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-pw 4134  df-sn 4151  df-pr 4153
This theorem is referenced by:  dis2ndc  21176  dislly  21213
  Copyright terms: Public domain W3C validator