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Theorem pred0 6327
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 6291 . 2 Pred(𝑅, ∅, 𝑋) = (∅ ∩ (𝑅 “ {𝑋}))
2 0in 4386 . 2 (∅ ∩ (𝑅 “ {𝑋})) = ∅
31, 2eqtri 2752 1 Pred(𝑅, ∅, 𝑋) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  cin 3940  c0 4315  {csn 4621  ccnv 5666  cima 5670  Predcpred 6290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-in 3948  df-nul 4316  df-pred 6291
This theorem is referenced by: (None)
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