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Theorem eqelsuc 4182
 Description: A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.)
Hypothesis
Ref Expression
eqelsuc.1 𝐴 ∈ V
Assertion
Ref Expression
eqelsuc (𝐴 = 𝐵𝐴 ∈ suc 𝐵)

Proof of Theorem eqelsuc
StepHypRef Expression
1 eqelsuc.1 . . 3 𝐴 ∈ V
21sucid 4180 . 2 𝐴 ∈ suc 𝐴
3 suceq 4165 . 2 (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
42, 3syl5eleq 2168 1 (𝐴 = 𝐵𝐴 ∈ suc 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1285   ∈ wcel 1434  Vcvv 2602  suc csuc 4128 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3412  df-suc 4134 This theorem is referenced by: (None)
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