Theorem eqsnm 3796

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 3795	. 2 |- ( E. x x e. A -> ( A C_ { B } <-> A = { B } ) )
2		dfss3 3182	. . 3 |- ( A C_ { B } <-> A. x e. A x e. { B } )
3		velsn 3650	. . . 4 |- ( x e. { B } <-> x = B )
4	3	ralbii 2512	. . 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 1373   E.wex 1515   e. wcel 2176   A.wral 2484   C_ wss 3166   {csn 3633

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-in 3172  df-ss 3179  df-sn 3639

This theorem is referenced by:  01eq0ring  13951  nninfall  15950