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Theorem pred0 6149
 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 6119 . 2 Pred(𝑅, ∅, 𝑋) = (∅ ∩ (𝑅 “ {𝑋}))
2 0in 4301 . 2 (∅ ∩ (𝑅 “ {𝑋})) = ∅
31, 2eqtri 2821 1 Pred(𝑅, ∅, 𝑋) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∩ cin 3880  ∅c0 4243  {csn 4525  ◡ccnv 5519   “ cima 5523  Predcpred 6118 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3443  df-dif 3884  df-in 3888  df-nul 4244  df-pred 6119 This theorem is referenced by:  trpred0  33224
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