Theorem sucid 4449

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 4448	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2164  Vcvv 2760  suc csuc 4397

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 3158  df-sn 3625  df-suc 4403

This theorem is referenced by:  eqelsuc  4451  unon  4544  ordunisuc2r  4547  ordsoexmid  4595  limom  4647  0elnn  4652  tfrexlem  6389  tfri1dALT  6406  tfrcl  6419  frecabcl  6454  phplem4  6913  fiintim  6987  fidcenumlemr  7016  nninfwlpoimlemginf  7237  pw1ne3  7292  sucpw1ne3  7294  sucpw1nel3  7295  prarloclemarch2  7481  prarloclemlt  7555  ennnfonelemex  12574  ennnfonelemrn  12579  bj-nn0suc0  15512  bj-nnelirr  15515  bj-inf2vnlem2  15533  bj-findis  15541  nninfsellemeq  15574