Theorem sucidALT 41212

Description: A set belongs to its successor. This proof was automatically derived from sucidALTVD 41211 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 4603	. . 3 ⊢ 𝐴 ∈ {𝐴}
3		elun1 4154	. . 3 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ ({𝐴} ∪ 𝐴))
4	2, 3	ax-mp 5	. 2 ⊢ 𝐴 ∈ ({𝐴} ∪ 𝐴)
5		df-suc 6199	. . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
6	5	equncomi 4133	. 2 ⊢ suc 𝐴 = ({𝐴} ∪ 𝐴)
7	4, 6	eleqtrri 2914	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2114  Vcvv 3496   ∪ cun 3936  {csn 4569  suc csuc 6195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-v 3498  df-un 3943  df-in 3945  df-ss 3954  df-sn 4570  df-suc 6199

This theorem is referenced by: (None)