Theorem sucidALT 42380

Description: A set belongs to its successor. This proof was automatically derived from sucidALTVD 42379 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 4594	. . 3 ⊢ 𝐴 ∈ {𝐴}
3		elun1 4106	. . 3 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ ({𝐴} ∪ 𝐴))
4	2, 3	ax-mp 5	. 2 ⊢ 𝐴 ∈ ({𝐴} ∪ 𝐴)
5		df-suc 6257	. . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
6	5	equncomi 4085	. 2 ⊢ suc 𝐴 = ({𝐴} ∪ 𝐴)
7	4, 6	eleqtrri 2838	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2108  Vcvv 3422   ∪ cun 3881  {csn 4558  suc csuc 6253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-un 3888  df-in 3890  df-ss 3900  df-sn 4559  df-suc 6257

This theorem is referenced by: (None)