Theorem sucprc 4167

Description: A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)

Assertion

Ref	Expression
sucprc	⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)

Proof of Theorem sucprc

Step	Hyp	Ref	Expression
1		df-suc 4126	. . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2		snprc 3457	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3		uneq2 3120	. . . 4 ⊢ ({𝐴} = ∅ → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
4	2, 3	sylbi 119	. . 3 ⊢ (¬ 𝐴 ∈ V → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
5	1, 4	syl5eq 2125	. 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = (𝐴 ∪ ∅))
6		un0 3278	. 2 ⊢ (𝐴 ∪ ∅) = 𝐴
7	5, 6	syl6eq 2129	1 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1284   ∈ wcel 1433  Vcvv 2601   ∪ cun 2971  ∅c0 3251  {csn 3398  suc csuc 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-un 2977  df-nul 3252  df-sn 3404  df-suc 4126

This theorem is referenced by:  sucprcreg  4292  sucon  4296