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  3783  snmg  3784  exmidsssnc  4286  fvunsng  5826  fsnunfv  5833  1stconst  6357  2ndconst  6358  tfr0dm  6458  tfrlemibxssdm  6463  tfrlemi14d  6469  tfr1onlembxssdm  6479  tfr1onlemres  6485  tfrcllembxssdm  6492  tfrcllemres  6498  en1uniel  6946  onunsnss  7067  snon0  7090  supsnti  7160  fseq1p1m1  10278  elfzomin  10399  swrds1  11186  fsumsplitsnun  11916  divalgmod  12424  setsslid  13069  bassetsnn  13075  1strbas  13136  srnginvld  13169  lmodvscad  13187  mgm1  13389  mnd1id  13475  0subm  13503  cnpdis  14901  upgr1edc  15906  bj-sels  16207