Theorem snsspw 2479

Description: The singleton of a class is a subset of its power class.

Assertion

Ref	Expression
snsspw	|- {A} (_ P~A

Proof of Theorem snsspw

Step	Hyp	Ref	Expression
1		eqimss 2109	. . 3 |- (x = A -> x (_ A)
2		elsn 2421	. . 3 |- (x e. {A} <-> x = A)
3		df-pw 2402	. . . 4 |- P~A = {x | x (_ A}
4	3	abeq2i 1570	. . 3 |- (x e. P~A <-> x (_ A)
5	1, 2, 4	3imtr4 219	. 2 |- (x e. {A} -> x e. P~A)
6	5	ssriv 2069	1 |- {A} (_ P~A

Syntax hints:   = wceq 956   e. wcel 958   (_ wss 2047  P~cpw 2401  {csn 2409

This theorem is referenced by:  snex 2750

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-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-in 2051  df-ss 2053  df-pw 2402  df-sn 2412

Copyright terms: Public domain