Theorem elsnc 3390

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 3389	. 2 ⊢ (A ∈ V → (A ∈ {B} ↔ A = B))
3	1, 2	ax-mp 7	1 ⊢ (A ∈ {B} ↔ A = B)

Syntax hints:   ↔ wb 98   = wceq 1242   ∈ wcel 1390  Vcvv 2551  {csn 3367

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sn 3373

This theorem is referenced by:  sneqr  3522  onsucelsucexmid  4215  ordsoexmid  4240  opthprc  4334  dmsnm  4729  dmsnopg  4735  cnvcnvsn  4740  sniota  4837  fsn  5278  eusvobj2  5441  opelreal  6726