Theorem eqsnm 3677

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		dfss3 3082	. . 3 |- ( A C_ { B } <-> A. x e. A x e. { B } )
2		velsn 3539	. . . 4 |- ( x e. { B } <-> x = B )
3	2	ralbii 2439	. . 3 |- ( A. x e. A x e. { B } <-> A. x e. A x = B )
4	1, 3	bitri 183	. 2 |- ( A C_ { B } <-> A. x e. A x = B )
5		sssnm 3676	. 2 |- ( E. x x e. A -> ( A C_ { B } <-> A = { B } ) )
6	4, 5	syl5rbbr 194	1 |- ( E. x x e. A -> ( A = { B } <-> A. x e. A x = B ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   E.wex 1468   e. wcel 1480   A.wral 2414   C_ wss 3066   {csn 3522

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-in 3072  df-ss 3079  df-sn 3528

This theorem is referenced by:  nninfall  13193