Theorem sucidALT 46113

Description: A set belongs to its successor. This proof was automatically derived from sucidALTVD 46112 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 4624	. . 3 ⊢ 𝐴 ∈ {𝐴}
3		elun1 4183	. . 3 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ ({𝐴} ∪ 𝐴))
4	2, 3	ax-mp 5	. 2 ⊢ 𝐴 ∈ ({𝐴} ∪ 𝐴)
5		df-suc 6394	. . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
6	5	equncomi 4162	. 2 ⊢ suc 𝐴 = ({𝐴} ∪ 𝐴)
7	4, 6	eleqtrri 2863	1 ⊢ 𝐴 ∈ suc 𝐴

This proof depends on syntax axioms:   ∈ wcel 2147  Vcvv 3454   ∪ cun 4152  {csn 4588  suc csuc 6390

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-un 4154  df-sn 4589  df-suc 6394

This theorem is used by: (None)