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Theorem pred0 6288
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 6252 . 2 Pred(𝑅, ∅, 𝑋) = (∅ ∩ (𝑅 “ {𝑋}))
2 0in 4352 . 2 (∅ ∩ (𝑅 “ {𝑋})) = ∅
31, 2eqtri 2764 1 Pred(𝑅, ∅, 𝑋) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  cin 3908  c0 4281  {csn 4585  ccnv 5631  cima 5635  Predcpred 6251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3407  df-v 3446  df-dif 3912  df-in 3916  df-nul 4282  df-pred 6252
This theorem is referenced by: (None)
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