Theorem bj-snglc 33608

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 2820	. 2 |- ( E. x e. B { A } = { x } <-> E. x ( x e. B /\ { A } = { x } ) )
2		bj-elsngl 33607	. 2 |- ( { A } e. sngl B <-> E. x e. B { A } = { x } )
3		elisset 3124	. . . . 5 |- ( A e. B -> E. x x = A )
4	3	pm4.71i 632	. . . 4 |- ( A e. B <-> ( A e. B /\ E. x x = A ) )
5		19.42v 1949	. . . 4 |- ( E. x ( A e. B /\ x = A ) <-> ( A e. B /\ E. x x = A ) )
6		eleq1 2539	. . . . . . 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 638	. . . . 5 |- ( ( A e. B /\ x = A ) <-> ( x e. B /\ x = A ) )
9	8	exbii 1644	. . . 4 |- ( E. x ( A e. B /\ x = A ) <-> E. x ( x e. B /\ x = A ) )
10	4, 5, 9	3bitr2i 273	. . 3 |- ( A e. B <-> E. x ( x e. B /\ x = A ) )
11		vex 3116	. . . . . . 7 |- x e. _V
12		sneqbg 4197	. . . . . . 7 |- ( x e. _V -> ( { x } = { A } <-> x = A ) )
13	11, 12	ax-mp 5	. . . . . 6 |- ( { x } = { A } <-> x = A )
14		eqcom 2476	. . . . . 6 |- ( { x } = { A } <-> { A } = { x } )
15	13, 14	bitr3i 251	. . . . 5 |- ( x = A <-> { A } = { x } )
16	15	anbi2i 694	. . . 4 |- ( ( x e. B /\ x = A ) <-> ( x e. B /\ { A } = { x } ) )
17	16	exbii 1644	. . 3 |- ( E. x ( x e. B /\ x = A ) <-> E. x ( x e. B /\ { A } = { x } ) )
18	10, 17	bitri 249	. 2 |- ( A e. B <-> E. x ( x e. B /\ { A } = { x } ) )
19	1, 2, 18	3bitr4ri 278	1 |- ( A e. B <-> { A } e. sngl B )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1379   E.wex 1596   e. wcel 1767   E.wrex 2815   _Vcvv 3113   {csn 4027  sngl bj-csngl 33604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-v 3115  df-dif 3479  df-un 3481  df-nul 3786  df-sn 4028  df-pr 4030  df-bj-sngl 33605

This theorem is referenced by:  bj-snglinv  33611  bj-tagci  33623  bj-tagcg  33624