Theorem mosn 3611

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 3693 and snprc 3640. (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 2900	. 2 |- E* x x = A
2		velsn 3592	. . 3 |- ( x e. { A } <-> x = A )
3	2	mobii 2051	. 2 |- ( E* x x e. { A } <-> E* x x = A )
4	1, 3	mpbir 145	1 |- E* x x e. { A }

Syntax hints:   = wceq 1343   E*wmo 2015   e. wcel 2136   {csn 3575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-v 2727  df-sn 3581

This theorem is referenced by: (None)