MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sucprc Structured version   Visualization version   GIF version

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
StepHypRef Expression
1 snprc 4285 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
21biimpi 206 . . 3 𝐴 ∈ V → {𝐴} = ∅)
32uneq2d 3800 . 2 𝐴 ∈ V → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
4 df-suc 5767 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
5 un0 4000 . . 3 (𝐴 ∪ ∅) = 𝐴
65eqcomi 2660 . 2 𝐴 = (𝐴 ∪ ∅)
73, 4, 63eqtr4g 2710 1 𝐴 ∈ V → suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
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
  Copyright terms: Public domain W3C validator