Theorem sucprc 6460

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		snprc 4717	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2	1	biimpi 216	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
3	2	uneq2d 4168	. 2 ⊢ (¬ 𝐴 ∈ V → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
4		df-suc 6390	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5		un0 4394	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
6	5	eqcomi 2746	. 2 ⊢ 𝐴 = (𝐴 ∪ ∅)
7	3, 4, 6	3eqtr4g 2802	1 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3480   ∪ cun 3949  ∅c0 4333  {csn 4626  suc csuc 6386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-un 3956  df-nul 4334  df-sn 4627  df-suc 6390

This theorem is referenced by:  nsuceq0  6467  sucon  7823  ordsuc  7833  ordsucOLD  7834  sucprcreg  9641  suc11reg  9659