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  7090  snon0  7113  supsnti  7183  fseq1p1m1  10302  elfzomin  10424  swrds1  11216  fsumsplitsnun  11946  divalgmod  12454  setsslid  13099  bassetsnn  13105  1strbas  13166  srnginvld  13199  lmodvscad  13217  mgm1  13419  mnd1id  13505  0subm  13533  cnpdis  14932  upgr1edc  15937  vtxd0nedgbfi  16059  wlk1walkdom  16105  bj-sels  16360