Theorem snidg 3605

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 2165	. 2 ⊢ 𝐴 = 𝐴
2		elsng 3591	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐴} ↔ 𝐴 = 𝐴))
3	1, 2	mpbiri 167	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})

Syntax hints:   → wi 4   = wceq 1343   ∈ wcel 2136  {csn 3576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sn 3582

This theorem is referenced by:  snidb  3606  elsn2g  3609  snnzg  3693  snmg  3694  exmidsssnc  4182  fvunsng  5679  fsnunfv  5686  1stconst  6189  2ndconst  6190  tfr0dm  6290  tfrlemibxssdm  6295  tfrlemi14d  6301  tfr1onlembxssdm  6311  tfr1onlemres  6317  tfrcllembxssdm  6324  tfrcllemres  6330  en1uniel  6770  onunsnss  6882  snon0  6901  supsnti  6970  fseq1p1m1  10029  elfzomin  10141  fsumsplitsnun  11360  divalgmod  11864  setsslid  12444  1strbas  12494  srnginvld  12521  lmodvscad  12532  mgm1  12601  cnpdis  12882  bj-sels  13796