Theorem sucid 4562

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 4561	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

This proof depends on syntax axioms:   ∈ wcel 2209  Vcvv 2821  suc csuc 4510

This proof depends on 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 proof 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 3715  df-suc 4516

This theorem is used by:  eqelsuc  4564  unon  4658  ordunisuc2r  4661  ordsoexmid  4709  limom  4761  0elnn  4766  tfrexlem  6605  tfri1dALT  6622  tfrcl  6635  frecabcl  6670  phplem4  7156  fiintim  7238  fidcenumlemr  7272  nninfwlpoimlemginf  7516  pw1ne3  7589  sucpw1ne3  7591  sucpw1nel3  7592  prarloclemarch2  7786  prarloclemlt  7860  ennnfonelemex  13307  ennnfonelemrn  13312  bj-nn0suc0  16988  bj-nnelirr  16991  bj-inf2vnlem2  17009  bj-findis  17017  3dom  17030  nninfsellemeq  17069