Theorem snid 3704

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 3703	. 2 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
3	1, 2	mpbi 145	1 ⊢ 𝐴 ∈ {𝐴}

Syntax hints:   ∈ wcel 2202  Vcvv 2803  {csn 3673

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-sn 3679

This theorem is referenced by:  vsnid  3705  exsnrex  3715  rabsnt  3750  sneqr  3848  undifexmid  4289  exmidexmid  4292  ss1o0el1  4293  exmidundif  4302  exmidundifim  4303  exmid1stab  4304  unipw  4315  intid  4322  ordtriexmidlem2  4624  ordtriexmid  4625  ontriexmidim  4626  ordtri2orexmid  4627  regexmidlem1  4637  0elsucexmid  4669  ordpwsucexmid  4674  opthprc  4783  fsn  5827  fsn2  5829  fvsn  5857  fvsnun1  5859  acexmidlema  6019  acexmidlemb  6020  acexmidlemab  6022  brtpos0  6461  mapsn  6902  mapsncnv  6907  0elixp  6941  en1  7016  djulclr  7291  djurclr  7292  djulcl  7293  djurcl  7294  djuf1olem  7295  exmidonfinlem  7447  elreal2  8093  1exp  10874  hashinfuni  11083  wrdexb  11172  0bits  12581  ennnfonelemhom  13097  dvef  15518  wlkl1loop  16279  djucllem  16498  bj-d0clsepcl  16621