Theorem snidg 3501

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 2100	. 2 ⊢ 𝐴 = 𝐴
2		elsng 3489	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐴} ↔ 𝐴 = 𝐴))
3	1, 2	mpbiri 167	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})

Syntax hints:   → wi 4   = wceq 1299   ∈ wcel 1448  {csn 3474

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-sn 3480

This theorem is referenced by:  snidb  3502  elsn2g  3505  snnzg  3587  snmg  3588  exmidsssnc  4064  fvunsng  5546  fsnunfv  5553  1stconst  6048  2ndconst  6049  tfr0dm  6149  tfrlemibxssdm  6154  tfrlemi14d  6160  tfr1onlembxssdm  6170  tfr1onlemres  6176  tfrcllembxssdm  6183  tfrcllemres  6189  en1uniel  6628  onunsnss  6734  snon0  6752  supsnti  6807  fseq1p1m1  9715  elfzomin  9824  fsumsplitsnun  11027  divalgmod  11419  setsslid  11791  1strbas  11840  srnginvld  11867  lmodvscad  11878  cnpdis  12192  bj-sels  12693