Theorem sucid 4244

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 4243	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 7	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 1438  Vcvv 2619  suc csuc 4192

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-sn 3452  df-suc 4198

This theorem is referenced by:  eqelsuc  4246  unon  4328  ordunisuc2r  4331  ordsoexmid  4378  limom  4428  0elnn  4432  tfrexlem  6099  tfri1dALT  6116  tfrcl  6129  frecabcl  6164  phplem4  6571  fiintim  6639  fidcenumlemr  6664  infnninf  6805  nnnninf  6806  prarloclemarch2  6978  prarloclemlt  7052  bj-nn0suc0  11845  bj-nnelirr  11848  bj-inf2vnlem2  11866  bj-findis  11874  nninfsellemeq  11906