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

Theorem sucprc 6433
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 4716 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
21biimpi 215 . . 3 𝐴 ∈ V → {𝐴} = ∅)
32uneq2d 4158 . 2 𝐴 ∈ V → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
4 df-suc 6363 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
5 un0 4385 . . 3 (𝐴 ∪ ∅) = 𝐴
65eqcomi 2735 . 2 𝐴 = (𝐴 ∪ ∅)
73, 4, 63eqtr4g 2791 1 𝐴 ∈ V → suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1533  wcel 2098  Vcvv 3468  cun 3941  c0 4317  {csn 4623  suc csuc 6359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-dif 3946  df-un 3948  df-nul 4318  df-sn 4624  df-suc 6363
This theorem is referenced by:  nsuceq0  6440  sucon  7787  ordsuc  7797  ordsucOLD  7798  sucprcreg  9595  suc11reg  9613
  Copyright terms: Public domain W3C validator