Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  insucid Structured version   Visualization version   GIF version

Theorem insucid 43393
Description: The intersection of a class and its successor is itself. (Contributed by RP, 3-Jan-2025.)
Assertion
Ref Expression
insucid (𝐴 ∩ suc 𝐴) = 𝐴

Proof of Theorem insucid
StepHypRef Expression
1 sssucid 6466 . 2 𝐴 ⊆ suc 𝐴
2 dfss2 3981 . 2 (𝐴 ⊆ suc 𝐴 ↔ (𝐴 ∩ suc 𝐴) = 𝐴)
31, 2mpbi 230 1 (𝐴 ∩ suc 𝐴) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  cin 3962  wss 3963  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-in 3970  df-ss 3980  df-suc 6392
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator