Theorem sucid 4377

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 4376	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2128  Vcvv 2712  suc csuc 4325

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3566  df-suc 4331

This theorem is referenced by:  eqelsuc  4379  unon  4470  ordunisuc2r  4473  ordsoexmid  4521  limom  4573  0elnn  4578  tfrexlem  6281  tfri1dALT  6298  tfrcl  6311  frecabcl  6346  phplem4  6800  fiintim  6873  fidcenumlemr  6899  pw1ne3  7165  sucpw1ne3  7167  sucpw1nel3  7168  prarloclemarch2  7339  prarloclemlt  7413  ennnfonelemex  12154  ennnfonelemrn  12159  bj-nn0suc0  13536  bj-nnelirr  13539  bj-inf2vnlem2  13557  bj-findis  13565  nninfsellemeq  13597