Theorem sucid 4543

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 4542	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2205  Vcvv 2815  suc csuc 4491

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-sn 3700  df-suc 4497

This theorem is referenced by:  eqelsuc  4545  unon  4638  ordunisuc2r  4641  ordsoexmid  4689  limom  4741  0elnn  4746  tfrexlem  6578  tfri1dALT  6595  tfrcl  6608  frecabcl  6643  phplem4  7122  fiintim  7204  fidcenumlemr  7238  nninfwlpoimlemginf  7480  pw1ne3  7553  sucpw1ne3  7555  sucpw1nel3  7556  prarloclemarch2  7750  prarloclemlt  7824  ennnfonelemex  13249  ennnfonelemrn  13254  bj-nn0suc0  16832  bj-nnelirr  16835  bj-inf2vnlem2  16853  bj-findis  16861  3dom  16874  nninfsellemeq  16904