Theorem sucprc 5838

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 4285	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2	1	biimpi 206	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
3	2	uneq2d 3800	. 2 ⊢ (¬ 𝐴 ∈ V → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
4		df-suc 5767	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5		un0 4000	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
6	5	eqcomi 2660	. 2 ⊢ 𝐴 = (𝐴 ∪ ∅)
7	3, 4, 6	3eqtr4g 2710	1 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∪ cun 3605  ∅c0 3948  {csn 4210  suc csuc 5763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-un 3612  df-nul 3949  df-sn 4211  df-suc 5767

This theorem is referenced by:  nsuceq0  5843  sucon  7050  ordsuc  7056  sucprcreg  8544  suc11reg  8554