Theorem snidg 3648

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 2193	. 2 ⊢ 𝐴 = 𝐴
2		elsng 3634	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐴} ↔ 𝐴 = 𝐴))
3	1, 2	mpbiri 168	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2164  {csn 3619

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-sn 3625

This theorem is referenced by:  snidb  3649  elsn2g  3652  snnzg  3736  snmg  3737  exmidsssnc  4233  fvunsng  5753  fsnunfv  5760  1stconst  6276  2ndconst  6277  tfr0dm  6377  tfrlemibxssdm  6382  tfrlemi14d  6388  tfr1onlembxssdm  6398  tfr1onlemres  6404  tfrcllembxssdm  6411  tfrcllemres  6417  en1uniel  6860  onunsnss  6975  snon0  6996  supsnti  7066  fseq1p1m1  10163  elfzomin  10276  fsumsplitsnun  11565  divalgmod  12071  setsslid  12672  1strbas  12738  srnginvld  12770  lmodvscad  12788  mgm1  12956  mnd1id  13031  0subm  13059  cnpdis  14421  bj-sels  15476