Theorem bj-elsngl 32956

Description: Characterization of the elements of the singletonization of a class. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-elsngl	|- ( A e. sngl B <-> E. x e. B A = { x } )

Distinct variable groups:   x, A   x, B

Proof of Theorem bj-elsngl

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clel 2618	. 2 |- ( A e. sngl B <-> E. y ( y = A /\ y e. sngl B ) )
2		df-bj-sngl 32954	. . . . 5 |- sngl B = { y | E. x e. B y = { x } }
3	2	abeq2i 2735	. . . 4 |- ( y e. sngl B <-> E. x e. B y = { x } )
4	3	anbi2i 730	. . 3 |- ( ( y = A /\ y e. sngl B ) <-> ( y = A /\ E. x e. B y = { x } ) )
5	4	exbii 1774	. 2 |- ( E. y ( y = A /\ y e. sngl B ) <-> E. y ( y = A /\ E. x e. B y = { x } ) )
6		r19.42v 3092	. . . . 5 |- ( E. x e. B ( y = A /\ y = { x } ) <-> ( y = A /\ E. x e. B y = { x } ) )
7	6	bicomi 214	. . . 4 |- ( ( y = A /\ E. x e. B y = { x } ) <-> E. x e. B ( y = A /\ y = { x } ) )
8	7	exbii 1774	. . 3 |- ( E. y ( y = A /\ E. x e. B y = { x } ) <-> E. y E. x e. B ( y = A /\ y = { x } ) )
9		rexcom4 3225	. . . 4 |- ( E. x e. B E. y ( y = A /\ y = { x } ) <-> E. y E. x e. B ( y = A /\ y = { x } ) )
10	9	bicomi 214	. . 3 |- ( E. y E. x e. B ( y = A /\ y = { x } ) <-> E. x e. B E. y ( y = A /\ y = { x } ) )
11		eqcom 2629	. . . . . 6 |- ( A = { x } <-> { x } = A )
12		snex 4908	. . . . . . 7 |- { x } e. _V
13	12	eqvinc 3330	. . . . . 6 |- ( { x } = A <-> E. y ( y = { x } /\ y = A ) )
14		exancom 1787	. . . . . 6 |- ( E. y ( y = { x } /\ y = A ) <-> E. y ( y = A /\ y = { x } ) )
15	11, 13, 14	3bitri 286	. . . . 5 |- ( A = { x } <-> E. y ( y = A /\ y = { x } ) )
16	15	bicomi 214	. . . 4 |- ( E. y ( y = A /\ y = { x } ) <-> A = { x } )
17	16	rexbii 3041	. . 3 |- ( E. x e. B E. y ( y = A /\ y = { x } ) <-> E. x e. B A = { x } )
18	8, 10, 17	3bitri 286	. 2 |- ( E. y ( y = A /\ E. x e. B y = { x } ) <-> E. x e. B A = { x } )
19	1, 5, 18	3bitri 286	1 |- ( A e. sngl B <-> E. x e. B A = { x } )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   {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-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-nul 3916  df-sn 4178  df-pr 4180  df-bj-sngl 32954

This theorem is referenced by:  bj-snglc  32957  bj-snglss  32958  bj-0nelsngl  32959  bj-eltag  32965