Theorem mosn 3741

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 3826 and snprc 3770. (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 3001	. 2 |- E* x x = A
2		velsn 3722	. . 3 |- ( x e. { A } <-> x = A )
3	2	mobii 2123	. 2 |- ( E* x x e. { A } <-> E* x x = A )
4	1, 3	mpbir 146	1 |- E* x x e. { A }

Syntax hints:   = wceq 1402   E*wmo 2087   e. wcel 2209   {csn 3705

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sn 3711

This theorem is referenced by: (None)