Theorem snsspw 1857

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 1548	. . 3 |- (x = A -> x (_ A)
2		elsn 1820	. . 3 |- (x e. {A} <-> x = A)
3		df-pw 1799	. . . 4 |- P~A = {x | x (_ A}
4	3	cleqabi 1176	. . 3 |- (x e. P~A <-> x (_ A)
5	1, 2, 4	3imtr4 192	. 2 |- (x e. {A} -> x e. P~A)
6	5	ssriv 1508	1 |- {A} (_ P~A

Syntax hints:   = wceq 1091   e. wcel 1092   (_ wss 1487  P~cpw 1798  {csn 1808

This theorem is referenced by:  snex 1859

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492  df-pw 1799  df-sn 1811

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