Theorem snidg 3664

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 2206	. 2 ⊢ 𝐴 = 𝐴
2		elsng 3650	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐴} ↔ 𝐴 = 𝐴))
3	1, 2	mpbiri 168	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2177  {csn 3635

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-sn 3641

This theorem is referenced by:  snidb  3665  elsn2g  3668  snnzg  3752  snmg  3753  exmidsssnc  4252  fvunsng  5788  fsnunfv  5795  1stconst  6317  2ndconst  6318  tfr0dm  6418  tfrlemibxssdm  6423  tfrlemi14d  6429  tfr1onlembxssdm  6439  tfr1onlemres  6445  tfrcllembxssdm  6452  tfrcllemres  6458  en1uniel  6906  onunsnss  7026  snon0  7049  supsnti  7119  fseq1p1m1  10229  elfzomin  10348  swrds1  11135  fsumsplitsnun  11780  divalgmod  12288  setsslid  12933  1strbas  12999  srnginvld  13032  lmodvscad  13050  mgm1  13252  mnd1id  13338  0subm  13366  cnpdis  14764  upgr1edc  15764  bj-sels  15964