Theorem exsnrex 3661

Description: There is a set being the element of a singleton if and only if there is an element of the singleton. (Contributed by Alexander van der Vekens, 1-Jan-2018.)

Assertion

Ref	Expression
exsnrex	|- ( E. x M = { x } <-> E. x e. M M = { x } )

Proof of Theorem exsnrex

Step	Hyp	Ref	Expression
1		vex 2763	. . . . . 6 |- x e. _V
2	1	snid 3650	. . . . 5 |- x e. { x }
3		eleq2 2257	. . . . 5 |- ( M = { x } -> ( x e. M <-> x e. { x } ) )
4	2, 3	mpbiri 168	. . . 4 |- ( M = { x } -> x e. M )
5	4	pm4.71ri 392	. . 3 |- ( M = { x } <-> ( x e. M /\ M = { x } ) )
6	5	exbii 1616	. 2 |- ( E. x M = { x } <-> E. x ( x e. M /\ M = { x } ) )
7		df-rex 2478	. 2 |- ( E. x e. M M = { x } <-> E. x ( x e. M /\ M = { x } ) )
8	6, 7	bitr4i 187	1 |- ( E. x M = { x } <-> E. x e. M M = { x } )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1503   e. wcel 2164   E.wrex 2473   {csn 3619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-sn 3625

This theorem is referenced by: (None)