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Theorem sucidALT 41978
 Description: A set belongs to its successor. This proof was automatically derived from sucidALTVD 41977 using translate_without_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
sucidALT.1 𝐴 ∈ V
Assertion
Ref Expression
sucidALT 𝐴 ∈ suc 𝐴

Proof of Theorem sucidALT
StepHypRef Expression
1 sucidALT.1 . . . 4 𝐴 ∈ V
21snid 4561 . . 3 𝐴 ∈ {𝐴}
3 elun1 4083 . . 3 (𝐴 ∈ {𝐴} → 𝐴 ∈ ({𝐴} ∪ 𝐴))
42, 3ax-mp 5 . 2 𝐴 ∈ ({𝐴} ∪ 𝐴)
5 df-suc 6179 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
65equncomi 4062 . 2 suc 𝐴 = ({𝐴} ∪ 𝐴)
74, 6eleqtrri 2851 1 𝐴 ∈ suc 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2111  Vcvv 3409   ∪ cun 3858  {csn 4525  suc csuc 6175 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-un 3865  df-in 3867  df-ss 3877  df-sn 4526  df-suc 6179 This theorem is referenced by: (None)
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