Theorem sucid 4452

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 4451	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2167  Vcvv 2763  suc csuc 4400

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-suc 4406

This theorem is referenced by:  eqelsuc  4454  unon  4547  ordunisuc2r  4550  ordsoexmid  4598  limom  4650  0elnn  4655  tfrexlem  6392  tfri1dALT  6409  tfrcl  6422  frecabcl  6457  phplem4  6916  fiintim  6992  fidcenumlemr  7021  nninfwlpoimlemginf  7242  pw1ne3  7297  sucpw1ne3  7299  sucpw1nel3  7300  prarloclemarch2  7486  prarloclemlt  7560  ennnfonelemex  12631  ennnfonelemrn  12636  bj-nn0suc0  15596  bj-nnelirr  15599  bj-inf2vnlem2  15617  bj-findis  15625  nninfsellemeq  15658