MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pred0 Structured version   Visualization version   GIF version

Theorem pred0 6355
Description: The predecessor class over is always . (Contributed by Scott Fenton, 16-Apr-2011.) (Proof shortened by AV, 11-Jun-2021.)
Assertion
Ref Expression
pred0 Pred(𝑅, ∅, 𝑋) = ∅

Proof of Theorem pred0
StepHypRef Expression
1 df-pred 6320 . 2 Pred(𝑅, ∅, 𝑋) = (∅ ∩ (𝑅 “ {𝑋}))
2 0in 4396 . 2 (∅ ∩ (𝑅 “ {𝑋})) = ∅
31, 2eqtri 2764 1 Pred(𝑅, ∅, 𝑋) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  cin 3949  c0 4332  {csn 4625  ccnv 5683  cima 5687  Predcpred 6319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-in 3957  df-nul 4333  df-pred 6320
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator