Theorem sucprc 6443

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 4674	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2	1	biimpi 219	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
3	2	uneq2d 4166	. 2 ⊢ (¬ 𝐴 ∈ V → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
4		df-suc 6370	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5		un0 4343	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
6	5	eqcomi 2769	. 2 ⊢ 𝐴 = (𝐴 ∪ ∅)
7	3, 4, 6	3eqtr4g 2820	1 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ∪ cun 4148  ∅c0 4277  {csn 4584  suc csuc 6366

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 4046  df-un 4150  df-nul 4278  df-sn 4585  df-suc 6370

This theorem is used by:  nsuceq0  6450  sucon  7819  ordsuc  7827  sucprcreg  9605  sucprcregOLD  9606  suc11reg  9625