Theorem snidg 3554

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 2139	. 2 ⊢ 𝐴 = 𝐴
2		elsng 3542	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐴} ↔ 𝐴 = 𝐴))
3	1, 2	mpbiri 167	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})

Syntax hints:   → wi 4   = wceq 1331   ∈ wcel 1480  {csn 3527

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sn 3533

This theorem is referenced by:  snidb  3555  elsn2g  3558  snnzg  3640  snmg  3641  exmidsssnc  4126  fvunsng  5614  fsnunfv  5621  1stconst  6118  2ndconst  6119  tfr0dm  6219  tfrlemibxssdm  6224  tfrlemi14d  6230  tfr1onlembxssdm  6240  tfr1onlemres  6246  tfrcllembxssdm  6253  tfrcllemres  6259  en1uniel  6698  onunsnss  6805  snon0  6824  supsnti  6892  fseq1p1m1  9874  elfzomin  9983  fsumsplitsnun  11188  divalgmod  11624  setsslid  12009  1strbas  12058  srnginvld  12085  lmodvscad  12096  cnpdis  12411  bj-sels  13112