Theorem snid 3697

Description: A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.)

Hypothesis

Ref	Expression
snid.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
snid	⊢ 𝐴 ∈ {𝐴}

Proof of Theorem snid

Step	Hyp	Ref	Expression
1		snid.1	. 2 ⊢ 𝐴 ∈ V
2		snidb 3696	. 2 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
3	1, 2	mpbi 145	1 ⊢ 𝐴 ∈ {𝐴}

Syntax hints:   ∈ wcel 2200  Vcvv 2799  {csn 3666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sn 3672

This theorem is referenced by:  vsnid  3698  exsnrex  3708  rabsnt  3741  sneqr  3837  undifexmid  4276  exmidexmid  4279  ss1o0el1  4280  exmidundif  4289  exmidundifim  4290  exmid1stab  4291  unipw  4302  intid  4309  ordtriexmidlem2  4609  ordtriexmid  4610  ontriexmidim  4611  ordtri2orexmid  4612  regexmidlem1  4622  0elsucexmid  4654  ordpwsucexmid  4659  opthprc  4767  fsn  5800  fsn2  5802  fvsn  5827  fvsnun1  5829  acexmidlema  5985  acexmidlemb  5986  acexmidlemab  5988  brtpos0  6388  mapsn  6827  mapsncnv  6832  0elixp  6866  en1  6941  djulclr  7204  djurclr  7205  djulcl  7206  djurcl  7207  djuf1olem  7208  exmidonfinlem  7359  elreal2  8005  1exp  10777  hashinfuni  10986  wrdexb  11070  0bits  12456  ennnfonelemhom  12972  dvef  15386  djucllem  16094  bj-d0clsepcl  16218