Theorem sucidALT 45562

Description: A set belongs to its successor. This proof was automatically derived from sucidALTVD 45561 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 4629	. . 3 ⊢ 𝐴 ∈ {𝐴}
3		elun1 4136	. . 3 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ ({𝐴} ∪ 𝐴))
4	2, 3	ax-mp 5	. 2 ⊢ 𝐴 ∈ ({𝐴} ∪ 𝐴)
5		df-suc 6368	. . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
6	5	equncomi 4115	. 2 ⊢ suc 𝐴 = ({𝐴} ∪ 𝐴)
7	4, 6	eleqtrri 2862	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2143  Vcvv 3455   ∪ cun 3904  {csn 4590  suc csuc 6364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-un 3911  df-ss 3923  df-sn 4591  df-suc 6368

This theorem is referenced by: (None)