Theorem sucid 4415

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 4414	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2148  Vcvv 2737  suc csuc 4363

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-suc 4369

This theorem is referenced by:  eqelsuc  4417  unon  4508  ordunisuc2r  4511  ordsoexmid  4559  limom  4611  0elnn  4616  tfrexlem  6330  tfri1dALT  6347  tfrcl  6360  frecabcl  6395  phplem4  6850  fiintim  6923  fidcenumlemr  6949  nninfwlpoimlemginf  7169  pw1ne3  7224  sucpw1ne3  7226  sucpw1nel3  7227  prarloclemarch2  7413  prarloclemlt  7487  ennnfonelemex  12405  ennnfonelemrn  12410  bj-nn0suc0  14473  bj-nnelirr  14476  bj-inf2vnlem2  14494  bj-findis  14502  nninfsellemeq  14534