Theorem sucid 4448

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 4447	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2164  Vcvv 2760  suc csuc 4396

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-suc 4402

This theorem is referenced by:  eqelsuc  4450  unon  4543  ordunisuc2r  4546  ordsoexmid  4594  limom  4646  0elnn  4651  tfrexlem  6387  tfri1dALT  6404  tfrcl  6417  frecabcl  6452  phplem4  6911  fiintim  6985  fidcenumlemr  7014  nninfwlpoimlemginf  7235  pw1ne3  7290  sucpw1ne3  7292  sucpw1nel3  7293  prarloclemarch2  7479  prarloclemlt  7553  ennnfonelemex  12571  ennnfonelemrn  12576  bj-nn0suc0  15442  bj-nnelirr  15445  bj-inf2vnlem2  15463  bj-findis  15471  nninfsellemeq  15504