Theorem bj-snglc 34875

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 2738	. 2 |- ( E. x e. B { A } = { x } <-> E. x ( x e. B /\ { A } = { x } ) )
2		bj-elsngl 34874	. 2 |- ( { A } e. sngl B <-> E. x e. B { A } = { x } )
3		elisset 3045	. . . . 5 |- ( A e. B -> E. x x = A )
4	3	pm4.71i 630	. . . 4 |- ( A e. B <-> ( A e. B /\ E. x x = A ) )
5		19.42v 1783	. . . 4 |- ( E. x ( A e. B /\ x = A ) <-> ( A e. B /\ E. x x = A ) )
6		eleq1 2454	. . . . . . 7 |- ( A = x -> ( A e. B <-> x e. B ) )
7	6	eqcoms 2394	. . . . . 6 |- ( x = A -> ( A e. B <-> x e. B ) )
8	7	pm5.32ri 636	. . . . 5 |- ( ( A e. B /\ x = A ) <-> ( x e. B /\ x = A ) )
9	8	exbii 1675	. . . 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 3037	. . . . . . 7 |- x e. _V
12		sneqbg 4114	. . . . . . 7 |- ( x e. _V -> ( { x } = { A } <-> x = A ) )
13	11, 12	ax-mp 5	. . . . . 6 |- ( { x } = { A } <-> x = A )
14		eqcom 2391	. . . . . 6 |- ( { x } = { A } <-> { A } = { x } )
15	13, 14	bitr3i 251	. . . . 5 |- ( x = A <-> { A } = { x } )
16	15	anbi2i 692	. . . 4 |- ( ( x e. B /\ x = A ) <-> ( x e. B /\ { A } = { x } ) )
17	16	exbii 1675	. . 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 367   = wceq 1399   E.wex 1620   e. wcel 1826   E.wrex 2733   _Vcvv 3034   {csn 3944  sngl bj-csngl 34871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-v 3036  df-dif 3392  df-un 3394  df-nul 3712  df-sn 3945  df-pr 3947  df-bj-sngl 34872

This theorem is referenced by:  bj-snglinv  34878  bj-tagci  34890  bj-tagcg  34891