Theorem pwsn 2500

Description: The power set of a singleton.

Assertion

Ref	Expression
pwsn	|- P~{A} = {(/), {A}}

Proof of Theorem pwsn

Step	Hyp	Ref	Expression
1		sssn 2473	. . 3 |- (x (_ {A} <-> (x = (/) \/ x = {A}))
2	1	abbii 1575	. 2 |- {x | x (_ {A}} = {x | (x = (/) \/ x = {A})}
3		df-pw 2402	. 2 |- P~{A} = {x | x (_ {A}}
4		dfpr2 2422	. 2 |- {(/), {A}} = {x | (x = (/) \/ x = {A})}
5	2, 3, 4	3eqtr4 1505	1 |- P~{A} = {(/), {A}}

Syntax hints:   \/ wo 222   = wceq 956  {cab 1463   (_ wss 2047  (/)c0 2280  P~cpw 2401  {csn 2409  {cpr 2410

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413

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