Theorem snidg 3612

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 2170	. 2 ⊢ 𝐴 = 𝐴
2		elsng 3598	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐴} ↔ 𝐴 = 𝐴))
3	1, 2	mpbiri 167	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})

Syntax hints:   → wi 4   = wceq 1348   ∈ wcel 2141  {csn 3583

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sn 3589

This theorem is referenced by:  snidb  3613  elsn2g  3616  snnzg  3700  snmg  3701  exmidsssnc  4189  fvunsng  5690  fsnunfv  5697  1stconst  6200  2ndconst  6201  tfr0dm  6301  tfrlemibxssdm  6306  tfrlemi14d  6312  tfr1onlembxssdm  6322  tfr1onlemres  6328  tfrcllembxssdm  6335  tfrcllemres  6341  en1uniel  6782  onunsnss  6894  snon0  6913  supsnti  6982  fseq1p1m1  10050  elfzomin  10162  fsumsplitsnun  11382  divalgmod  11886  setsslid  12466  1strbas  12517  srnginvld  12544  lmodvscad  12555  mgm1  12624  mnd1id  12680  0subm  12702  cnpdis  13036  bj-sels  13949