Theorem snidg 3561

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 2140	. 2 ⊢ 𝐴 = 𝐴
2		elsng 3547	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐴} ↔ 𝐴 = 𝐴))
3	1, 2	mpbiri 167	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})

Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481  {csn 3532

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sn 3538

This theorem is referenced by:  snidb  3562  elsn2g  3565  snnzg  3648  snmg  3649  exmidsssnc  4134  fvunsng  5622  fsnunfv  5629  1stconst  6126  2ndconst  6127  tfr0dm  6227  tfrlemibxssdm  6232  tfrlemi14d  6238  tfr1onlembxssdm  6248  tfr1onlemres  6254  tfrcllembxssdm  6261  tfrcllemres  6267  en1uniel  6706  onunsnss  6813  snon0  6832  supsnti  6900  fseq1p1m1  9905  elfzomin  10014  fsumsplitsnun  11220  divalgmod  11660  setsslid  12048  1strbas  12097  srnginvld  12124  lmodvscad  12135  cnpdis  12450  bj-sels  13283