Theorem sucid 4564

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 4563	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

This proof depends on syntax axioms:   ∈ wcel 2209  Vcvv 2821  suc csuc 4512

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3715  df-suc 4518

This theorem is used by:  eqelsuc  4566  unon  4660  ordunisuc2r  4663  ordsoexmid  4711  limom  4763  0elnn  4768  tfrexlem  6609  tfri1dALT  6626  tfrcl  6639  frecabcl  6674  phplem4  7161  fiintim  7243  fidcenumlemr  7278  nninfwlpoimlemginf  7523  pw1ne3  7596  sucpw1ne3  7598  sucpw1nel3  7599  prarloclemarch2  7793  prarloclemlt  7867  ennnfonelemex  13364  ennnfonelemrn  13369  bj-nn0suc0  17193  bj-nnelirr  17196  bj-inf2vnlem2  17214  bj-findis  17222  3dom  17235  nninfsellemeq  17275