Theorem mosn 3560

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 3641 and snprc 3588. (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 2859	. 2 |- E* x x = A
2		velsn 3544	. . 3 |- ( x e. { A } <-> x = A )
3	2	mobii 2036	. 2 |- ( E* x x e. { A } <-> E* x x = A )
4	1, 3	mpbir 145	1 |- E* x x e. { A }

Syntax hints:   = wceq 1331   e. wcel 1480   E*wmo 2000   {csn 3527

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sn 3533

This theorem is referenced by: (None)