Theorem sucid 4537

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 4536	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2203  Vcvv 2812  suc csuc 4485

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-un 3214  df-sn 3694  df-suc 4491

This theorem is referenced by:  eqelsuc  4539  unon  4632  ordunisuc2r  4635  ordsoexmid  4683  limom  4735  0elnn  4740  tfrexlem  6564  tfri1dALT  6581  tfrcl  6594  frecabcl  6629  phplem4  7108  fiintim  7190  fidcenumlemr  7224  nninfwlpoimlemginf  7466  pw1ne3  7539  sucpw1ne3  7541  sucpw1nel3  7542  prarloclemarch2  7733  prarloclemlt  7807  ennnfonelemex  13157  ennnfonelemrn  13162  bj-nn0suc0  16712  bj-nnelirr  16715  bj-inf2vnlem2  16733  bj-findis  16741  3dom  16754  nninfsellemeq  16784