Theorem snidg 3621

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 2177	. 2 ⊢ 𝐴 = 𝐴
2		elsng 3607	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐴} ↔ 𝐴 = 𝐴))
3	1, 2	mpbiri 168	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148  {csn 3592

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sn 3598

This theorem is referenced by:  snidb  3622  elsn2g  3625  snnzg  3709  snmg  3710  exmidsssnc  4201  fvunsng  5707  fsnunfv  5714  1stconst  6217  2ndconst  6218  tfr0dm  6318  tfrlemibxssdm  6323  tfrlemi14d  6329  tfr1onlembxssdm  6339  tfr1onlemres  6345  tfrcllembxssdm  6352  tfrcllemres  6358  en1uniel  6799  onunsnss  6911  snon0  6930  supsnti  6999  fseq1p1m1  10087  elfzomin  10199  fsumsplitsnun  11418  divalgmod  11922  setsslid  12503  1strbas  12566  srnginvld  12598  lmodvscad  12616  mgm1  12719  mnd1id  12776  0subm  12799  cnpdis  13524  bj-sels  14437