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Theorem pred0 6335
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 6299 . 2 Pred(𝑅, ∅, 𝑋) = (∅ ∩ (𝑅 “ {𝑋}))
2 0in 4389 . 2 (∅ ∩ (𝑅 “ {𝑋})) = ∅
31, 2eqtri 2756 1 Pred(𝑅, ∅, 𝑋) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  cin 3944  c0 4318  {csn 4624  ccnv 5671  cima 5675  Predcpred 6298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3429  df-v 3472  df-dif 3948  df-in 3952  df-nul 4319  df-pred 6299
This theorem is referenced by: (None)
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