Theorem eqsn 4361

Description: Two ways to express that a nonempty set equals a singleton. (Contributed by NM, 15-Dec-2007.) (Proof shortened by JJ, 23-Jul-2021.)

Assertion

Ref	Expression
eqsn	|- ( A =/= (/) -> ( A = { B } <-> A. x e. A x = B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem eqsn

Step	Hyp	Ref	Expression
1		df-ne 2795	. . 3 |- ( A =/= (/) <-> -. A = (/) )
2		biorf 420	. . 3 |- ( -. A = (/) -> ( A = { B } <-> ( A = (/) \/ A = { B } ) ) )
3	1, 2	sylbi 207	. 2 |- ( A =/= (/) -> ( A = { B } <-> ( A = (/) \/ A = { B } ) ) )
4		dfss3 3592	. . 3 |- ( A C_ { B } <-> A. x e. A x e. { B } )
5		sssn 4358	. . 3 |- ( A C_ { B } <-> ( A = (/) \/ A = { B } ) )
6		velsn 4193	. . . 4 |- ( x e. { B } <-> x = B )
7	6	ralbii 2980	. . 3 |- ( A. x e. A x e. { B } <-> A. x e. A x = B )
8	4, 5, 7	3bitr3i 290	. 2 |- ( ( A = (/) \/ A = { B } ) <-> A. x e. A x = B )
9	3, 8	syl6bb 276	1 |- ( A =/= (/) -> ( A = { B } <-> A. x e. A x = B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)c0 3915   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178

This theorem is referenced by:  issn  4363  zornn0g  9327  hashgt12el  13210  hashgt12el2  13211  hashge2el2dif  13262  lssne0  18951  qtopeu  21519  rngoueqz  33739  mapdm0OLD  39383  lmod0rng  41868