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  3838  undifexmid  4278  exmidexmid  4281  ss1o0el1  4282  exmidundif  4291  exmidundifim  4292  exmid1stab  4293  unipw  4304  intid  4311  ordtriexmidlem2  4613  ordtriexmid  4614  ontriexmidim  4615  ordtri2orexmid  4616  regexmidlem1  4626  0elsucexmid  4658  ordpwsucexmid  4663  opthprc  4772  fsn  5812  fsn2  5814  fvsn  5841  fvsnun1  5843  acexmidlema  6001  acexmidlemb  6002  acexmidlemab  6004  brtpos0  6409  mapsn  6850  mapsncnv  6855  0elixp  6889  en1  6964  djulclr  7232  djurclr  7233  djulcl  7234  djurcl  7235  djuf1olem  7236  exmidonfinlem  7387  elreal2  8033  1exp  10807  hashinfuni  11016  wrdexb  11101  0bits  12491  ennnfonelemhom  13007  dvef  15422  wlkl1loop  16130  djucllem  16273  bj-d0clsepcl  16397