Theorem sucid 4514

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 4513	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2202  Vcvv 2802  suc csuc 4462

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-suc 4468

This theorem is referenced by:  eqelsuc  4516  unon  4609  ordunisuc2r  4612  ordsoexmid  4660  limom  4712  0elnn  4717  tfrexlem  6500  tfri1dALT  6517  tfrcl  6530  frecabcl  6565  phplem4  7041  fiintim  7123  fidcenumlemr  7154  nninfwlpoimlemginf  7375  pw1ne3  7448  sucpw1ne3  7450  sucpw1nel3  7451  prarloclemarch2  7639  prarloclemlt  7713  ennnfonelemex  13040  ennnfonelemrn  13045  bj-nn0suc0  16571  bj-nnelirr  16574  bj-inf2vnlem2  16592  bj-findis  16600  3dom  16613  nninfsellemeq  16642