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Theorem sucid 4154
 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
StepHypRef Expression
1 sucid.1 . 2 𝐴 ∈ V
2 sucidg 4153 . 2 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
31, 2ax-mp 7 1 𝐴 ∈ suc 𝐴
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1393  Vcvv 2557  suc csuc 4102 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-suc 4108 This theorem is referenced by:  eqelsuc  4156  unon  4237  ordunisuc2r  4240  ordsoexmid  4286  limom  4336  0elnn  4340  tfrexlem  5948  phplem4  6318  prarloclemarch2  6517  prarloclemlt  6591  bj-nn0suc0  10075  bj-nnelirr  10078  bj-inf2vnlem2  10096  bj-findis  10104
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