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

Theorem sucprcreg 8544
Description: A class is equal to its successor iff it is a proper class (assuming the Axiom of Regularity). (Contributed by NM, 9-Jul-2004.) (Proof shortened by BJ, 16-Apr-2019.)
Assertion
Ref Expression
sucprcreg 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)

Proof of Theorem sucprcreg
StepHypRef Expression
1 sucprc 5838 . 2 𝐴 ∈ V → suc 𝐴 = 𝐴)
2 elirr 8543 . . . 4 ¬ 𝐴𝐴
3 df-suc 5767 . . . . . . . 8 suc 𝐴 = (𝐴 ∪ {𝐴})
43eqeq1i 2656 . . . . . . 7 (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
5 ssequn2 3819 . . . . . . 7 ({𝐴} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
64, 5bitr4i 267 . . . . . 6 (suc 𝐴 = 𝐴 ↔ {𝐴} ⊆ 𝐴)
76biimpi 206 . . . . 5 (suc 𝐴 = 𝐴 → {𝐴} ⊆ 𝐴)
8 snidg 4239 . . . . 5 (𝐴 ∈ V → 𝐴 ∈ {𝐴})
9 ssel2 3631 . . . . 5 (({𝐴} ⊆ 𝐴𝐴 ∈ {𝐴}) → 𝐴𝐴)
107, 8, 9syl2an 493 . . . 4 ((suc 𝐴 = 𝐴𝐴 ∈ V) → 𝐴𝐴)
112, 10mto 188 . . 3 ¬ (suc 𝐴 = 𝐴𝐴 ∈ V)
1211imnani 438 . 2 (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V)
131, 12impbii 199 1 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cun 3605  wss 3607  {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  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-reg 8538
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-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-pr 4213  df-suc 5767
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator