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  4277  exmidexmid  4280  ss1o0el1  4281  exmidundif  4290  exmidundifim  4291  exmid1stab  4292  unipw  4303  intid  4310  ordtriexmidlem2  4612  ordtriexmid  4613  ontriexmidim  4614  ordtri2orexmid  4615  regexmidlem1  4625  0elsucexmid  4657  ordpwsucexmid  4662  opthprc  4770  fsn  5809  fsn2  5811  fvsn  5838  fvsnun1  5840  acexmidlema  5998  acexmidlemb  5999  acexmidlemab  6001  brtpos0  6404  mapsn  6845  mapsncnv  6850  0elixp  6884  en1  6959  djulclr  7224  djurclr  7225  djulcl  7226  djurcl  7227  djuf1olem  7228  exmidonfinlem  7379  elreal2  8025  1exp  10798  hashinfuni  11007  wrdexb  11091  0bits  12478  ennnfonelemhom  12994  dvef  15409  wlkl1loop  16079  djucllem  16188  bj-d0clsepcl  16312