Theorem snid 3402

Description: A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.)

Hypothesis

Ref	Expression
snid.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
snid	⊢ 𝐴 ∈ {𝐴}

Proof of Theorem snid

Step	Hyp	Ref	Expression
1		snid.1	. 2 ⊢ 𝐴 ∈ V
2		snidb 3401	. 2 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
3	1, 2	mpbi 133	1 ⊢ 𝐴 ∈ {𝐴}

Syntax hints:   ∈ wcel 1393  Vcvv 2557  {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sn 3381

This theorem is referenced by:  vsnid  3403  exsnrex  3413  rabsnt  3445  sneqr  3531  rext  3951  unipw  3953  intid  3960  snnex  4181  ordtriexmidlem2  4246  ordtriexmid  4247  ordtri2orexmid  4248  regexmidlem1  4258  0elsucexmid  4289  ordpwsucexmid  4294  opthprc  4391  fsn  5335  fsn2  5337  fvsn  5358  fvsnun1  5360  acexmidlema  5503  acexmidlemb  5504  acexmidlemab  5506  brtpos0  5867  en1  6279  elreal2  6907  1exp  9284  bj-d0clsepcl  10049