Theorem mosn 3709

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 3794 and snprc 3738. (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 2982	. 2 |- E* x x = A
2		velsn 3690	. . 3 |- ( x e. { A } <-> x = A )
3	2	mobii 2116	. 2 |- ( E* x x e. { A } <-> E* x x = A )
4	1, 3	mpbir 146	1 |- E* x x e. { A }

Syntax hints:   = wceq 1398   E*wmo 2080   e. wcel 2202   {csn 3673

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-sn 3679

This theorem is referenced by: (None)