Theorem sucprc 6388

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 4649	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2	1	biimpi 217	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
3	2	uneq2d 4098	. 2 ⊢ (¬ 𝐴 ∈ V → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
4		df-suc 6316	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5		un0 4322	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
6	5	eqcomi 2748	. 2 ⊢ 𝐴 = (𝐴 ∪ ∅)
7	3, 4, 6	3eqtr4g 2799	1 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1547   ∈ wcel 2119  Vcvv 3431   ∪ cun 3881  ∅c0 4261  {csn 4555  suc csuc 6312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-dif 3886  df-un 3888  df-nul 4262  df-sn 4556  df-suc 6316

This theorem is referenced by:  nsuceq0  6395  sucon  7746  ordsuc  7754  sucprcreg  9511  sucprcregOLD  9512  suc11reg  9531