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 4678	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2	1	biimpi 219	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
3	2	uneq2d 4115	. 2 ⊢ (¬ 𝐴 ∈ V → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
4		df-suc 6367	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5		un0 4344	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
6	5	eqcomi 2770	. 2 ⊢ 𝐴 = (𝐴 ∪ ∅)
7	3, 4, 6	3eqtr4g 2821	1 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ∪ cun 3897  ∅c0 4279  {csn 4584  suc csuc 6363

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-dif 3902  df-un 3904  df-nul 4280  df-sn 4585  df-suc 6367

This theorem is used by:  nsuceq0  6447  sucon  7815  ordsuc  7823  sucprcreg  9593  sucprcregOLD  9594  suc11reg  9613