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Theorem predel 5884
Description: Membership in the predecessor class implies membership in the base class. (Contributed by Scott Fenton, 11-Feb-2011.)
Assertion
Ref Expression
predel (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝐴)

Proof of Theorem predel
StepHypRef Expression
1 elinel1 3963 . 2 (𝑌 ∈ (𝐴 ∩ (𝑅 “ {𝑋})) → 𝑌𝐴)
2 df-pred 5867 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
31, 2eleq2s 2862 1 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2155  cin 3733  {csn 4336  ccnv 5278  cima 5282  Predcpred 5866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-in 3741  df-pred 5867
This theorem is referenced by:  predpo  5885  predpoirr  5895  predfrirr  5896  dftrpred3g  32197
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