Theorem sucid 4418

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 4417	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2148  Vcvv 2738  suc csuc 4366

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 2740  df-un 3134  df-sn 3599  df-suc 4372

This theorem is referenced by:  eqelsuc  4420  unon  4511  ordunisuc2r  4514  ordsoexmid  4562  limom  4614  0elnn  4619  tfrexlem  6335  tfri1dALT  6352  tfrcl  6365  frecabcl  6400  phplem4  6855  fiintim  6928  fidcenumlemr  6954  nninfwlpoimlemginf  7174  pw1ne3  7229  sucpw1ne3  7231  sucpw1nel3  7232  prarloclemarch2  7418  prarloclemlt  7492  ennnfonelemex  12415  ennnfonelemrn  12420  bj-nn0suc0  14705  bj-nnelirr  14708  bj-inf2vnlem2  14726  bj-findis  14734  nninfsellemeq  14766