Theorem sucid 4419

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 4418	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2148  Vcvv 2739  suc csuc 4367

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-sn 3600  df-suc 4373

This theorem is referenced by:  eqelsuc  4421  unon  4512  ordunisuc2r  4515  ordsoexmid  4563  limom  4615  0elnn  4620  tfrexlem  6337  tfri1dALT  6354  tfrcl  6367  frecabcl  6402  phplem4  6857  fiintim  6930  fidcenumlemr  6956  nninfwlpoimlemginf  7176  pw1ne3  7231  sucpw1ne3  7233  sucpw1nel3  7234  prarloclemarch2  7420  prarloclemlt  7494  ennnfonelemex  12417  ennnfonelemrn  12422  bj-nn0suc0  14787  bj-nnelirr  14790  bj-inf2vnlem2  14808  bj-findis  14816  nninfsellemeq  14848