Theorem elsingles 32025

Description: Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)

Assertion

Ref	Expression
elsingles	|- ( A e. Singletons <-> E. x A = { x } )

Distinct variable group:   x, A

Proof of Theorem elsingles

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( A e. Singletons -> A e. _V )
2		snex 4908	. . . 4 |- { x } e. _V
3		eleq1 2689	. . . 4 |- ( A = { x } -> ( A e. _V <-> { x } e. _V ) )
4	2, 3	mpbiri 248	. . 3 |- ( A = { x } -> A e. _V )
5	4	exlimiv 1858	. 2 |- ( E. x A = { x } -> A e. _V )
6		eleq1 2689	. . 3 |- ( y = A -> ( y e. Singletons <-> A e. Singletons ) )
7		eqeq1 2626	. . . 4 |- ( y = A -> ( y = { x } <-> A = { x } ) )
8	7	exbidv 1850	. . 3 |- ( y = A -> ( E. x y = { x } <-> E. x A = { x } ) )
9		df-singles 31970	. . . . 5 |- Singletons = ran Singleton
10	9	eleq2i 2693	. . . 4 |- ( y e. Singletons <-> y e. ran Singleton )
11		vex 3203	. . . . 5 |- y e. _V
12	11	elrn 5366	. . . 4 |- ( y e. ran Singleton <-> E. x xSingleton y )
13		vex 3203	. . . . . 6 |- x e. _V
14	13, 11	brsingle 32024	. . . . 5 |- ( xSingleton y <-> y = { x } )
15	14	exbii 1774	. . . 4 |- ( E. x xSingleton y <-> E. x y = { x } )
16	10, 12, 15	3bitri 286	. . 3 |- ( y e. Singletons <-> E. x y = { x } )
17	6, 8, 16	vtoclbg 3267	. 2 |- ( A e. _V -> ( A e. Singletons <-> E. x A = { x } ) )
18	1, 5, 17	pm5.21nii 368	1 |- ( A e. Singletons <-> E. x A = { x } )

Syntax hints:   <-> wb 196   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   {csn 4177   class class class wbr 4653   ran crn 5115  Singletoncsingle 31945   Singletonscsingles 31946

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-symdif 3844  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-eprel 5029  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169  df-txp 31961  df-singleton 31969  df-singles 31970

This theorem is referenced by:  dfsingles2  32028  snelsingles  32029  funpartlem  32049