Theorem eqsnm 3547

Description: Two ways to express that an inhabited set equals a singleton. (Contributed by Jim Kingdon, 11-Aug-2018.)

Assertion

Ref	Expression
eqsnm	|- ( E. x x e. A -> ( A = { B } <-> A. x e. A x = B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem eqsnm

Step	Hyp	Ref	Expression
1		dfss3 2989	. . 3 |- ( A C_ { B } <-> A. x e. A x e. { B } )
2		velsn 3415	. . . 4 |- ( x e. { B } <-> x = B )
3	2	ralbii 2372	. . 3 |- ( A. x e. A x e. { B } <-> A. x e. A x = B )
4	1, 3	bitri 182	. 2 |- ( A C_ { B } <-> A. x e. A x = B )
5		sssnm 3546	. 2 |- ( E. x x e. A -> ( A C_ { B } <-> A = { B } ) )
6	4, 5	syl5rbbr 193	1 |- ( E. x x e. A -> ( A = { B } <-> A. x e. A x = B ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   A.wral 2348   C_ wss 2973   {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-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-in 2979  df-ss 2986  df-sn 3404

This theorem is referenced by: (None)