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Theorem sucidALT 43617
Description: A set belongs to its successor. This proof was automatically derived from sucidALTVD 43616 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 4663 . . 3 𝐴 ∈ {𝐴}
3 elun1 4175 . . 3 (𝐴 ∈ {𝐴} → 𝐴 ∈ ({𝐴} ∪ 𝐴))
42, 3ax-mp 5 . 2 𝐴 ∈ ({𝐴} ∪ 𝐴)
5 df-suc 6367 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
65equncomi 4154 . 2 suc 𝐴 = ({𝐴} ∪ 𝐴)
74, 6eleqtrri 2832 1 𝐴 ∈ suc 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  Vcvv 3474  cun 3945  {csn 4627  suc csuc 6363
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-un 3952  df-in 3954  df-ss 3964  df-sn 4628  df-suc 6367
This theorem is referenced by: (None)
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