Theorem sucid 4505

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 4504	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2200  Vcvv 2799  suc csuc 4453

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-suc 4459

This theorem is referenced by:  eqelsuc  4507  unon  4600  ordunisuc2r  4603  ordsoexmid  4651  limom  4703  0elnn  4708  tfrexlem  6470  tfri1dALT  6487  tfrcl  6500  frecabcl  6535  phplem4  7004  fiintim  7081  fidcenumlemr  7110  nninfwlpoimlemginf  7331  pw1ne3  7403  sucpw1ne3  7405  sucpw1nel3  7406  prarloclemarch2  7594  prarloclemlt  7668  ennnfonelemex  12971  ennnfonelemrn  12976  bj-nn0suc0  16243  bj-nnelirr  16246  bj-inf2vnlem2  16264  bj-findis  16272  nninfsellemeq  16311