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  4287  fvunsng  5837  fsnunfv  5844  1stconst  6373  2ndconst  6374  tfr0dm  6474  tfrlemibxssdm  6479  tfrlemi14d  6485  tfr1onlembxssdm  6495  tfr1onlemres  6501  tfrcllembxssdm  6508  tfrcllemres  6514  en1uniel  6964  onunsnss  7087  snon0  7110  supsnti  7180  fseq1p1m1  10298  elfzomin  10420  swrds1  11208  fsumsplitsnun  11938  divalgmod  12446  setsslid  13091  bassetsnn  13097  1strbas  13158  srnginvld  13191  lmodvscad  13209  mgm1  13411  mnd1id  13497  0subm  13525  cnpdis  14924  upgr1edc  15929  wlk1walkdom  16080  bj-sels  16301