Theorem elsnc 3373

Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)

Hypothesis

Ref	Expression
elsnc.1	⊢ A ∈ V

Assertion

Ref	Expression
elsnc	⊢ (A ∈ {B} ↔ A = B)

Proof of Theorem elsnc

Step	Hyp	Ref	Expression
1		elsnc.1	. 2 ⊢ A ∈ V
2		elsncg 3372	. 2 ⊢ (A ∈ V → (A ∈ {B} ↔ A = B))
3	1, 2	ax-mp 7	1 ⊢ (A ∈ {B} ↔ A = B)

Syntax hints:   ↔ wb 98   = wceq 1228   ∈ wcel 1374  Vcvv 2535  {csn 3350

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-sn 3356

This theorem is referenced by:  sneqr  3505  onsucelsucexmid  4199  ordsoexmid  4224  opthprc  4318  dmsnm  4713  dmsnopg  4719  cnvcnvsn  4724  sniota  4821  fsn  5260  eusvobj2  5422  opelreal  6540