Theorem bj-snglc 31633

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 2762	. 2 |- ( E. x e. B { A } = { x } <-> E. x ( x e. B /\ { A } = { x } ) )
2		bj-elsngl 31632	. 2 |- ( { A } e. sngl B <-> E. x e. B { A } = { x } )
3		elisset 3043	. . . . 5 |- ( A e. B -> E. x x = A )
4	3	pm4.71i 644	. . . 4 |- ( A e. B <-> ( A e. B /\ E. x x = A ) )
5		19.42v 1842	. . . 4 |- ( E. x ( A e. B /\ x = A ) <-> ( A e. B /\ E. x x = A ) )
6		eleq1 2537	. . . . . . 7 |- ( A = x -> ( A e. B <-> x e. B ) )
7	6	eqcoms 2479	. . . . . 6 |- ( x = A -> ( A e. B <-> x e. B ) )
8	7	pm5.32ri 650	. . . . 5 |- ( ( A e. B /\ x = A ) <-> ( x e. B /\ x = A ) )
9	8	exbii 1726	. . . 4 |- ( E. x ( A e. B /\ x = A ) <-> E. x ( x e. B /\ x = A ) )
10	4, 5, 9	3bitr2i 281	. . 3 |- ( A e. B <-> E. x ( x e. B /\ x = A ) )
11		vex 3034	. . . . . . 7 |- x e. _V
12		sneqbg 4134	. . . . . . 7 |- ( x e. _V -> ( { x } = { A } <-> x = A ) )
13	11, 12	ax-mp 5	. . . . . 6 |- ( { x } = { A } <-> x = A )
14		eqcom 2478	. . . . . 6 |- ( { x } = { A } <-> { A } = { x } )
15	13, 14	bitr3i 259	. . . . 5 |- ( x = A <-> { A } = { x } )
16	15	anbi2i 708	. . . 4 |- ( ( x e. B /\ x = A ) <-> ( x e. B /\ { A } = { x } ) )
17	16	exbii 1726	. . 3 |- ( E. x ( x e. B /\ x = A ) <-> E. x ( x e. B /\ { A } = { x } ) )
18	10, 17	bitri 257	. 2 |- ( A e. B <-> E. x ( x e. B /\ { A } = { x } ) )
19	1, 2, 18	3bitr4ri 286	1 |- ( A e. B <-> { A } e. sngl B )

Syntax hints:   <-> wb 189   /\ wa 376   = wceq 1452   E.wex 1671   e. wcel 1904   E.wrex 2757   _Vcvv 3031   {csn 3959  sngl bj-csngl 31629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-v 3033  df-dif 3393  df-un 3395  df-nul 3723  df-sn 3960  df-pr 3962  df-bj-sngl 31630

This theorem is referenced by:  bj-snglinv  31636  bj-tagci  31648  bj-tagcg  31649