Theorem snid 3719

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 3718	. 2 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
3	1, 2	mpbi 145	1 ⊢ 𝐴 ∈ {𝐴}

Syntax hints:   ∈ wcel 2203  Vcvv 2812  {csn 3688

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-sn 3694

This theorem is referenced by:  vsnid  3720  exsnrex  3730  rabsnt  3765  sneqr  3863  undifexmid  4305  exmidexmid  4308  ss1o0el1  4309  exmidundif  4318  exmidundifim  4319  exmid1stab  4320  unipw  4332  intid  4339  ordtriexmidlem2  4641  ordtriexmid  4642  ontriexmidim  4643  ordtri2orexmid  4644  regexmidlem1  4654  0elsucexmid  4686  ordpwsucexmid  4691  opthprc  4800  fsn  5848  fsn2  5850  fvsn  5878  fvsnun1  5880  acexmidlema  6040  acexmidlemb  6041  acexmidlemab  6043  brtpos0  6482  mapsn  6924  mapsncnv  6929  0elixp  6963  en1  7038  djulclr  7339  djurclr  7340  djulcl  7341  djurcl  7342  djuf1olem  7343  exmidonfinlem  7495  elreal2  8144  1exp  10929  hashinfuni  11138  wrdexb  11232  0bits  12641  ennnfonelemhom  13158  dvef  15584  wlkl1loop  16345  djucllem  16564  bj-d0clsepcl  16687