Theorem sucid 4509

Description: A set belongs to its successor. (Contributed by NM, 22-Jun-1994.) (Proof shortened by Alan Sare, 18-Feb-2012.) (Proof shortened by Scott Fenton, 20-Feb-2012.)

Hypothesis

Ref	Expression
sucid.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
sucid	⊢ 𝐴 ∈ suc 𝐴

Proof of Theorem sucid

Step	Hyp	Ref	Expression
1		sucid.1	. 2 ⊢ 𝐴 ∈ V
2		sucidg 4508	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2200  Vcvv 2799  suc csuc 4457

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-suc 4463

This theorem is referenced by:  eqelsuc  4511  unon  4604  ordunisuc2r  4607  ordsoexmid  4655  limom  4707  0elnn  4712  tfrexlem  6491  tfri1dALT  6508  tfrcl  6521  frecabcl  6556  phplem4  7029  fiintim  7109  fidcenumlemr  7138  nninfwlpoimlemginf  7359  pw1ne3  7431  sucpw1ne3  7433  sucpw1nel3  7434  prarloclemarch2  7622  prarloclemlt  7696  ennnfonelemex  13006  ennnfonelemrn  13011  bj-nn0suc0  16422  bj-nnelirr  16425  bj-inf2vnlem2  16443  bj-findis  16451  3dom  16465  nninfsellemeq  16494