Theorem sucid 4402

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 4401	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2141  Vcvv 2730  suc csuc 4350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-suc 4356

This theorem is referenced by:  eqelsuc  4404  unon  4495  ordunisuc2r  4498  ordsoexmid  4546  limom  4598  0elnn  4603  tfrexlem  6313  tfri1dALT  6330  tfrcl  6343  frecabcl  6378  phplem4  6833  fiintim  6906  fidcenumlemr  6932  nninfwlpoimlemginf  7152  pw1ne3  7207  sucpw1ne3  7209  sucpw1nel3  7210  prarloclemarch2  7381  prarloclemlt  7455  ennnfonelemex  12369  ennnfonelemrn  12374  bj-nn0suc0  13985  bj-nnelirr  13988  bj-inf2vnlem2  14006  bj-findis  14014  nninfsellemeq  14047