Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sucidALT Structured version   Visualization version   GIF version

Theorem sucidALT 44869
Description: A set belongs to its successor. This proof was automatically derived from sucidALTVD 44868 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 4667 . . 3 𝐴 ∈ {𝐴}
3 elun1 4192 . . 3 (𝐴 ∈ {𝐴} → 𝐴 ∈ ({𝐴} ∪ 𝐴))
42, 3ax-mp 5 . 2 𝐴 ∈ ({𝐴} ∪ 𝐴)
5 df-suc 6392 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
65equncomi 4170 . 2 suc 𝐴 = ({𝐴} ∪ 𝐴)
74, 6eleqtrri 2838 1 𝐴 ∈ suc 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  Vcvv 3478  cun 3961  {csn 4631  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-ss 3980  df-sn 4632  df-suc 6392
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator