Theorem eqsnm 3843

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 3842	. 2 |- ( E. x x e. A -> ( A C_ { B } <-> A = { B } ) )
2		dfss3 3217	. . 3 |- ( A C_ { B } <-> A. x e. A x e. { B } )
3		velsn 3690	. . . 4 |- ( x e. { B } <-> x = B )
4	3	ralbii 2539	. . 3 |- ( A. x e. A x e. { B } <-> A. x e. A x = B )
5	2, 4	bitri 184	. 2 |- ( A C_ { B } <-> A. x e. A x = B )
6	1, 5	bitr3di 195	1 |- ( E. x x e. A -> ( A = { B } <-> A. x e. A x = B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2202   A.wral 2511   C_ wss 3201   {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-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-in 3207  df-ss 3214  df-sn 3679

This theorem is referenced by:  01eq0ring  14284  nninfall  16735