Theorem sucid 4469

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 4468	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2177  Vcvv 2773  suc csuc 4417

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3172  df-sn 3641  df-suc 4423

This theorem is referenced by:  eqelsuc  4471  unon  4564  ordunisuc2r  4567  ordsoexmid  4615  limom  4667  0elnn  4672  tfrexlem  6430  tfri1dALT  6447  tfrcl  6460  frecabcl  6495  phplem4  6964  fiintim  7040  fidcenumlemr  7069  nninfwlpoimlemginf  7290  pw1ne3  7355  sucpw1ne3  7357  sucpw1nel3  7358  prarloclemarch2  7545  prarloclemlt  7619  ennnfonelemex  12835  ennnfonelemrn  12840  bj-nn0suc0  16000  bj-nnelirr  16003  bj-inf2vnlem2  16021  bj-findis  16029  nninfsellemeq  16066