Theorem snidg 3695

Description: A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.)

Assertion

Ref	Expression
snidg	⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})

Proof of Theorem snidg

Step	Hyp	Ref	Expression
1		eqid 2229	. 2 ⊢ 𝐴 = 𝐴
2		elsng 3681	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐴} ↔ 𝐴 = 𝐴))
3	1, 2	mpbiri 168	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  {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:  snidb  3696  elsn2g  3699  snnzg  3784  snmg  3785  exmidsssnc  4288  fvunsng  5840  fsnunfv  5847  1stconst  6378  2ndconst  6379  tfr0dm  6479  tfrlemibxssdm  6484  tfrlemi14d  6490  tfr1onlembxssdm  6500  tfr1onlemres  6506  tfrcllembxssdm  6513  tfrcllemres  6519  en1uniel  6969  onunsnss  7095  snon0  7118  supsnti  7188  fseq1p1m1  10307  elfzomin  10429  swrds1  11221  fsumsplitsnun  11951  divalgmod  12459  setsslid  13104  bassetsnn  13110  1strbas  13171  srnginvld  13204  lmodvscad  13222  mgm1  13424  mnd1id  13510  0subm  13538  cnpdis  14937  upgr1edc  15942  uspgr1edc  16059  vtxd0nedgbfi  16085  wlk1walkdom  16131  bj-sels  16386