Theorem eqsnm 3833

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 3832	. 2 |- ( E. x x e. A -> ( A C_ { B } <-> A = { B } ) )
2		dfss3 3213	. . 3 |- ( A C_ { B } <-> A. x e. A x e. { B } )
3		velsn 3683	. . . 4 |- ( x e. { B } <-> x = B )
4	3	ralbii 2536	. . 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 1395   E.wex 1538   e. wcel 2200   A.wral 2508   C_ wss 3197   {csn 3666

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-in 3203  df-ss 3210  df-sn 3672

This theorem is referenced by:  01eq0ring  14153  nninfall  16375