Theorem eqsnm 3526

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 2935	. . 3 |- ( A C_ { B } <-> A. x e. A x e. { B } )
2		velsn 3392	. . . 4 |- ( x e. { B } <-> x = B )
3	2	ralbii 2330	. . 3 |- ( A. x e. A x e. { B } <-> A. x e. A x = B )
4	1, 3	bitri 173	. 2 |- ( A C_ { B } <-> A. x e. A x = B )
5		sssnm 3525	. 2 |- ( E. x x e. A -> ( A C_ { B } <-> A = { B } ) )
6	4, 5	syl5rbbr 184	1 |- ( E. x x e. A -> ( A = { B } <-> A. x e. A x = B ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   A.wral 2306   C_ wss 2917   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-sn 3381

This theorem is referenced by: (None)