Theorem eqsnm 3742

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		sssnm 3741	. 2 |- ( E. x x e. A -> ( A C_ { B } <-> A = { B } ) )
2		dfss3 3137	. . 3 |- ( A C_ { B } <-> A. x e. A x e. { B } )
3		velsn 3600	. . . 4 |- ( x e. { B } <-> x = B )
4	3	ralbii 2476	. . 3 |- ( A. x e. A x e. { B } <-> A. x e. A x = B )
5	2, 4	bitri 183	. 2 |- ( A C_ { B } <-> A. x e. A x = B )
6	1, 5	bitr3di 194	1 |- ( E. x x e. A -> ( A = { B } <-> A. x e. A x = B ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   E.wex 1485   e. wcel 2141   A.wral 2448   C_ wss 3121   {csn 3583

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-sn 3589

This theorem is referenced by:  nninfall  14042