 New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  pwsn GIF version

Theorem pwsn 3881
 Description: The power set of a singleton. (Contributed by NM, 5-Jun-2006.)
Assertion
Ref Expression
pwsn {A} = {, {A}}

Proof of Theorem pwsn
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 sssn 3864 . . 3 (x {A} ↔ (x = x = {A}))
21abbii 2465 . 2 {x x {A}} = {x (x = x = {A})}
3 df-pw 3724 . 2 {A} = {x x {A}}
4 dfpr2 3749 . 2 {, {A}} = {x (x = x = {A})}
52, 3, 43eqtr4i 2383 1 {A} = {, {A}}
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 357   = wceq 1642  {cab 2339   ⊆ wss 3257  ∅c0 3550  ℘cpw 3722  {csn 3737  {cpr 3738 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-pr 3742 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator