Theorem sucid 4508

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 4507	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2200  Vcvv 2799  suc csuc 4456

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 4462

This theorem is referenced by:  eqelsuc  4510  unon  4603  ordunisuc2r  4606  ordsoexmid  4654  limom  4706  0elnn  4711  tfrexlem  6486  tfri1dALT  6503  tfrcl  6516  frecabcl  6551  phplem4  7024  fiintim  7101  fidcenumlemr  7130  nninfwlpoimlemginf  7351  pw1ne3  7423  sucpw1ne3  7425  sucpw1nel3  7426  prarloclemarch2  7614  prarloclemlt  7688  ennnfonelemex  12993  ennnfonelemrn  12998  bj-nn0suc0  16337  bj-nnelirr  16340  bj-inf2vnlem2  16358  bj-findis  16366  3dom  16381  nninfsellemeq  16410