Theorem sucid 4395

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 4394	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2136  Vcvv 2726  suc csuc 4343

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-suc 4349

This theorem is referenced by:  eqelsuc  4397  unon  4488  ordunisuc2r  4491  ordsoexmid  4539  limom  4591  0elnn  4596  tfrexlem  6302  tfri1dALT  6319  tfrcl  6332  frecabcl  6367  phplem4  6821  fiintim  6894  fidcenumlemr  6920  pw1ne3  7186  sucpw1ne3  7188  sucpw1nel3  7189  prarloclemarch2  7360  prarloclemlt  7434  ennnfonelemex  12347  ennnfonelemrn  12352  bj-nn0suc0  13832  bj-nnelirr  13835  bj-inf2vnlem2  13853  bj-findis  13861  nninfsellemeq  13894