Theorem sucid 4417

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 4416	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2148  Vcvv 2737  suc csuc 4365

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 4371

This theorem is referenced by:  eqelsuc  4419  unon  4510  ordunisuc2r  4513  ordsoexmid  4561  limom  4613  0elnn  4618  tfrexlem  6334  tfri1dALT  6351  tfrcl  6364  frecabcl  6399  phplem4  6854  fiintim  6927  fidcenumlemr  6953  nninfwlpoimlemginf  7173  pw1ne3  7228  sucpw1ne3  7230  sucpw1nel3  7231  prarloclemarch2  7417  prarloclemlt  7491  ennnfonelemex  12414  ennnfonelemrn  12419  bj-nn0suc0  14672  bj-nnelirr  14675  bj-inf2vnlem2  14693  bj-findis  14701  nninfsellemeq  14733