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Theorem eqelsuc 5989
 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 5987 . 2 𝐴 ∈ suc 𝐴
3 suceq 5973 . 2 (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
42, 3syl5eleq 2850 1 (𝐴 = 𝐵𝐴 ∈ suc 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1652   ∈ wcel 2155  Vcvv 3350  suc csuc 5910 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743 This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-un 3737  df-sn 4335  df-suc 5914 This theorem is referenced by:  pssnn  8385
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