Theorem mosn 3429

Description: A singleton has at most one element. This works whether A is a proper class or not, and in that sense can be seen as encompassing both snmg 3508 and snprc 3457. (Contributed by Jim Kingdon, 30-Aug-2018.)

Assertion

Ref	Expression
mosn	|- E* x x e. { A }

Distinct variable group:   x, A

Proof of Theorem mosn

Step	Hyp	Ref	Expression
1		moeq 2767	. 2 |- E* x x = A
2		velsn 3415	. . 3 |- ( x e. { A } <-> x = A )
3	2	mobii 1978	. 2 |- ( E* x x e. { A } <-> E* x x = A )
4	1, 3	mpbir 144	1 |- E* x x e. { A }

Syntax hints:   = wceq 1284   e. wcel 1433   E*wmo 1942   {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sn 3404

This theorem is referenced by: (None)