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Theorem sucidALT 38129
 Description: A set belongs to its successor. This proof was automatically derived from sucidALTVD 38128 using translatewithout_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 4155 . . 3 𝐴 ∈ {𝐴}
3 elun1 3742 . . 3 (𝐴 ∈ {𝐴} → 𝐴 ∈ ({𝐴} ∪ 𝐴))
42, 3ax-mp 5 . 2 𝐴 ∈ ({𝐴} ∪ 𝐴)
5 df-suc 5646 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
65equncomi 3721 . 2 suc 𝐴 = ({𝐴} ∪ 𝐴)
74, 6eleqtrri 2687 1 𝐴 ∈ suc 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538  {csn 4125  suc csuc 5642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-in 3547  df-ss 3554  df-sn 4126  df-suc 5646 This theorem is referenced by: (None)
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