Theorem snidg 3651

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 2196	. 2 ⊢ 𝐴 = 𝐴
2		elsng 3637	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐴} ↔ 𝐴 = 𝐴))
3	1, 2	mpbiri 168	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167  {csn 3622

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sn 3628

This theorem is referenced by:  snidb  3652  elsn2g  3655  snnzg  3739  snmg  3740  exmidsssnc  4236  fvunsng  5756  fsnunfv  5763  1stconst  6279  2ndconst  6280  tfr0dm  6380  tfrlemibxssdm  6385  tfrlemi14d  6391  tfr1onlembxssdm  6401  tfr1onlemres  6407  tfrcllembxssdm  6414  tfrcllemres  6420  en1uniel  6863  onunsnss  6978  snon0  7001  supsnti  7071  fseq1p1m1  10169  elfzomin  10282  fsumsplitsnun  11584  divalgmod  12092  setsslid  12729  1strbas  12795  srnginvld  12827  lmodvscad  12845  mgm1  13013  mnd1id  13088  0subm  13116  cnpdis  14478  bj-sels  15560