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Theorem pred0 6053
 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 6023 . 2 Pred(𝑅, ∅, 𝑋) = (∅ ∩ (𝑅 “ {𝑋}))
2 0in 4267 . 2 (∅ ∩ (𝑅 “ {𝑋})) = ∅
31, 2eqtri 2819 1 Pred(𝑅, ∅, 𝑋) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1522   ∩ cin 3858  ∅c0 4211  {csn 4472  ◡ccnv 5442   “ cima 5446  Predcpred 6022 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-v 3439  df-dif 3862  df-in 3866  df-nul 4212  df-pred 6023 This theorem is referenced by:  trpred0  32685
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