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

Theorem pred0 5853
 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 5823 . 2 Pred(𝑅, ∅, 𝑋) = (∅ ∩ (𝑅 “ {𝑋}))
2 0in 4113 . 2 (∅ ∩ (𝑅 “ {𝑋})) = ∅
31, 2eqtri 2793 1 Pred(𝑅, ∅, 𝑋) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1631   ∩ cin 3722  ∅c0 4063  {csn 4316  ◡ccnv 5248   “ cima 5252  Predcpred 5822 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-dif 3726  df-in 3730  df-nul 4064  df-pred 5823 This theorem is referenced by:  trpred0  32072
 Copyright terms: Public domain W3C validator