Theorem eqsnm 3773

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 3772	. 2 |- ( E. x x e. A -> ( A C_ { B } <-> A = { B } ) )
2		dfss3 3160	. . 3 |- ( A C_ { B } <-> A. x e. A x e. { B } )
3		velsn 3627	. . . 4 |- ( x e. { B } <-> x = B )
4	3	ralbii 2496	. . 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 1364   E.wex 1503   e. wcel 2160   A.wral 2468   C_ wss 3144   {csn 3610

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-in 3150  df-ss 3157  df-sn 3616

This theorem is referenced by:  01eq0ring  13561  nninfall  15245