Theorem bj-snglc 31531

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 2777	. 2 |- ( E. x e. B { A } = { x } <-> E. x ( x e. B /\ { A } = { x } ) )
2		bj-elsngl 31530	. 2 |- ( { A } e. sngl B <-> E. x e. B { A } = { x } )
3		elisset 3091	. . . . 5 |- ( A e. B -> E. x x = A )
4	3	pm4.71i 636	. . . 4 |- ( A e. B <-> ( A e. B /\ E. x x = A ) )
5		19.42v 1827	. . . 4 |- ( E. x ( A e. B /\ x = A ) <-> ( A e. B /\ E. x x = A ) )
6		eleq1 2495	. . . . . . 7 |- ( A = x -> ( A e. B <-> x e. B ) )
7	6	eqcoms 2434	. . . . . 6 |- ( x = A -> ( A e. B <-> x e. B ) )
8	7	pm5.32ri 642	. . . . 5 |- ( ( A e. B /\ x = A ) <-> ( x e. B /\ x = A ) )
9	8	exbii 1712	. . . 4 |- ( E. x ( A e. B /\ x = A ) <-> E. x ( x e. B /\ x = A ) )
10	4, 5, 9	3bitr2i 276	. . 3 |- ( A e. B <-> E. x ( x e. B /\ x = A ) )
11		vex 3083	. . . . . . 7 |- x e. _V
12		sneqbg 4170	. . . . . . 7 |- ( x e. _V -> ( { x } = { A } <-> x = A ) )
13	11, 12	ax-mp 5	. . . . . 6 |- ( { x } = { A } <-> x = A )
14		eqcom 2431	. . . . . 6 |- ( { x } = { A } <-> { A } = { x } )
15	13, 14	bitr3i 254	. . . . 5 |- ( x = A <-> { A } = { x } )
16	15	anbi2i 698	. . . 4 |- ( ( x e. B /\ x = A ) <-> ( x e. B /\ { A } = { x } ) )
17	16	exbii 1712	. . 3 |- ( E. x ( x e. B /\ x = A ) <-> E. x ( x e. B /\ { A } = { x } ) )
18	10, 17	bitri 252	. 2 |- ( A e. B <-> E. x ( x e. B /\ { A } = { x } ) )
19	1, 2, 18	3bitr4ri 281	1 |- ( A e. B <-> { A } e. sngl B )

Syntax hints:   <-> wb 187   /\ wa 370   = wceq 1437   E.wex 1657   e. wcel 1872   E.wrex 2772   _Vcvv 3080   {csn 3998  sngl bj-csngl 31527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-v 3082  df-dif 3439  df-un 3441  df-nul 3762  df-sn 3999  df-pr 4001  df-bj-sngl 31528

This theorem is referenced by:  bj-snglinv  31534  bj-tagci  31546  bj-tagcg  31547