Theorem sucid 4560

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 4559	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2209  Vcvv 2821  suc csuc 4508

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3714  df-suc 4514

This theorem is referenced by:  eqelsuc  4562  unon  4656  ordunisuc2r  4659  ordsoexmid  4707  limom  4759  0elnn  4764  tfrexlem  6599  tfri1dALT  6616  tfrcl  6629  frecabcl  6664  phplem4  7150  fiintim  7232  fidcenumlemr  7266  nninfwlpoimlemginf  7510  pw1ne3  7583  sucpw1ne3  7585  sucpw1nel3  7586  prarloclemarch2  7780  prarloclemlt  7854  ennnfonelemex  13288  ennnfonelemrn  13293  bj-nn0suc0  16959  bj-nnelirr  16962  bj-inf2vnlem2  16980  bj-findis  16988  3dom  17001  nninfsellemeq  17031