Theorem sucprc 6440

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 4697	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2	1	biimpi 216	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
3	2	uneq2d 4148	. 2 ⊢ (¬ 𝐴 ∈ V → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
4		df-suc 6369	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5		un0 4374	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
6	5	eqcomi 2743	. 2 ⊢ 𝐴 = (𝐴 ∪ ∅)
7	3, 4, 6	3eqtr4g 2794	1 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ∈ wcel 2107  Vcvv 3463   ∪ cun 3929  ∅c0 4313  {csn 4606  suc csuc 6365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3465  df-dif 3934  df-un 3936  df-nul 4314  df-sn 4607  df-suc 6369

This theorem is referenced by:  nsuceq0  6447  sucon  7805  ordsuc  7815  ordsucOLD  7816  sucprcreg  9623  suc11reg  9641