Theorem sucidALT 45464

Description: A set belongs to its successor. This proof was automatically derived from sucidALTVD 45463 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

Step	Hyp	Ref	Expression
1		sucidALT.1	. . . 4 ⊢ 𝐴 ∈ V
2	1	snid 4630	. . 3 ⊢ 𝐴 ∈ {𝐴}
3		elun1 4143	. . 3 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ ({𝐴} ∪ 𝐴))
4	2, 3	ax-mp 5	. 2 ⊢ 𝐴 ∈ ({𝐴} ∪ 𝐴)
5		df-suc 6363	. . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
6	5	equncomi 4122	. 2 ⊢ suc 𝐴 = ({𝐴} ∪ 𝐴)
7	4, 6	eleqtrri 2868	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2149  Vcvv 3463   ∪ cun 3911  {csn 4591  suc csuc 6359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-un 3918  df-ss 3930  df-sn 4592  df-suc 6363

This theorem is referenced by: (None)