Theorem sucid 4520

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 4519	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2202  Vcvv 2803  suc csuc 4468

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-suc 4474

This theorem is referenced by:  eqelsuc  4522  unon  4615  ordunisuc2r  4618  ordsoexmid  4666  limom  4718  0elnn  4723  tfrexlem  6543  tfri1dALT  6560  tfrcl  6573  frecabcl  6608  phplem4  7084  fiintim  7166  fidcenumlemr  7197  nninfwlpoimlemginf  7418  pw1ne3  7491  sucpw1ne3  7493  sucpw1nel3  7494  prarloclemarch2  7682  prarloclemlt  7756  ennnfonelemex  13096  ennnfonelemrn  13101  bj-nn0suc0  16646  bj-nnelirr  16649  bj-inf2vnlem2  16667  bj-findis  16675  3dom  16688  nninfsellemeq  16720