Theorem bj-snglc 32957

Description: Characterization of the elements of A in terms of elements of its singletonization. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-snglc	|- ( A e. B <-> { A } e. sngl B )

Proof of Theorem bj-snglc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rex 2918	. 2 |- ( E. x e. B { A } = { x } <-> E. x ( x e. B /\ { A } = { x } ) )
2		bj-elsngl 32956	. 2 |- ( { A } e. sngl B <-> E. x e. B { A } = { x } )
3		elisset 3215	. . . . 5 |- ( A e. B -> E. x x = A )
4	3	pm4.71i 664	. . . 4 |- ( A e. B <-> ( A e. B /\ E. x x = A ) )
5		19.42v 1918	. . . 4 |- ( E. x ( A e. B /\ x = A ) <-> ( A e. B /\ E. x x = A ) )
6		eleq1 2689	. . . . . . 7 |- ( A = x -> ( A e. B <-> x e. B ) )
7	6	eqcoms 2630	. . . . . 6 |- ( x = A -> ( A e. B <-> x e. B ) )
8	7	pm5.32ri 670	. . . . 5 |- ( ( A e. B /\ x = A ) <-> ( x e. B /\ x = A ) )
9	8	exbii 1774	. . . 4 |- ( E. x ( A e. B /\ x = A ) <-> E. x ( x e. B /\ x = A ) )
10	4, 5, 9	3bitr2i 288	. . 3 |- ( A e. B <-> E. x ( x e. B /\ x = A ) )
11		vex 3203	. . . . . . 7 |- x e. _V
12		sneqbg 4374	. . . . . . 7 |- ( x e. _V -> ( { x } = { A } <-> x = A ) )
13	11, 12	ax-mp 5	. . . . . 6 |- ( { x } = { A } <-> x = A )
14		eqcom 2629	. . . . . 6 |- ( { x } = { A } <-> { A } = { x } )
15	13, 14	bitr3i 266	. . . . 5 |- ( x = A <-> { A } = { x } )
16	15	anbi2i 730	. . . 4 |- ( ( x e. B /\ x = A ) <-> ( x e. B /\ { A } = { x } ) )
17	16	exbii 1774	. . 3 |- ( E. x ( x e. B /\ x = A ) <-> E. x ( x e. B /\ { A } = { x } ) )
18	10, 17	bitri 264	. 2 |- ( A e. B <-> E. x ( x e. B /\ { A } = { x } ) )
19	1, 2, 18	3bitr4ri 293	1 |- ( A e. B <-> { A } e. sngl B )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   {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-snglinv  32960  bj-tagci  32972  bj-tagcg  32973