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Theorem sucidALT 39421
Description: A set belongs to its successor. This proof was automatically derived from sucidALTVD 39420 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 4241 . . 3 𝐴 ∈ {𝐴}
3 elun1 3813 . . 3 (𝐴 ∈ {𝐴} → 𝐴 ∈ ({𝐴} ∪ 𝐴))
42, 3ax-mp 5 . 2 𝐴 ∈ ({𝐴} ∪ 𝐴)
5 df-suc 5767 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
65equncomi 3792 . 2 suc 𝐴 = ({𝐴} ∪ 𝐴)
74, 6eleqtrri 2729 1 𝐴 ∈ suc 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2030  Vcvv 3231  cun 3605  {csn 4210  suc csuc 5763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-in 3614  df-ss 3621  df-sn 4211  df-suc 5767
This theorem is referenced by: (None)
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