Theorem bj-snglss 32958

Description: The singletonization of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-snglss	|- sngl A C_ ~P A

Proof of Theorem bj-snglss

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-elsngl 32956	. . . . 5 |- ( x e. sngl A <-> E. y e. A x = { y } )
2		df-rex 2918	. . . . . 6 |- ( E. y e. A x = { y } <-> E. y ( y e. A /\ x = { y } ) )
3		snssi 4339	. . . . . . . 8 |- ( y e. A -> { y } C_ A )
4		sseq1 3626	. . . . . . . . 9 |- ( x = { y } -> ( x C_ A <-> { y } C_ A ) )
5	4	biimparc 504	. . . . . . . 8 |- ( ( { y } C_ A /\ x = { y } ) -> x C_ A )
6	3, 5	sylan 488	. . . . . . 7 |- ( ( y e. A /\ x = { y } ) -> x C_ A )
7	6	eximi 1762	. . . . . 6 |- ( E. y ( y e. A /\ x = { y } ) -> E. y x C_ A )
8	2, 7	sylbi 207	. . . . 5 |- ( E. y e. A x = { y } -> E. y x C_ A )
9	1, 8	sylbi 207	. . . 4 |- ( x e. sngl A -> E. y x C_ A )
10		ax5e 1841	. . . 4 |- ( E. y x C_ A -> x C_ A )
11	9, 10	syl 17	. . 3 |- ( x e. sngl A -> x C_ A )
12		selpw 4165	. . 3 |- ( x e. ~P A <-> x C_ A )
13	11, 12	sylibr 224	. 2 |- ( x e. sngl A -> x e. ~P A )
14	13	ssriv 3607	1 |- sngl A C_ ~P A

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   C_ wss 3574   ~Pcpw 4158   {csn 4177  sngl bj-csngl 32953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-bj-sngl 32954

This theorem is referenced by:  bj-snglex  32961  bj-tagss  32968