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

Theorem pred0 5895
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 5865 . 2 Pred(𝑅, ∅, 𝑋) = (∅ ∩ (𝑅 “ {𝑋}))
2 0in 4131 . 2 (∅ ∩ (𝑅 “ {𝑋})) = ∅
31, 2eqtri 2787 1 Pred(𝑅, ∅, 𝑋) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  cin 3731  c0 4079  {csn 4334  ccnv 5276  cima 5280  Predcpred 5864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-dif 3735  df-in 3739  df-nul 4080  df-pred 5865
This theorem is referenced by:  trpred0  32111
  Copyright terms: Public domain W3C validator