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Theorem predel 6259
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 4142 . 2 (𝑌 ∈ (𝐴 ∩ (𝑅 “ {𝑋})) → 𝑌𝐴)
2 df-pred 6238 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
31, 2eleq2s 2855 1 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  cin 3897  {csn 4573  ccnv 5619  cima 5623  Predcpred 6237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443  df-in 3905  df-pred 6238
This theorem is referenced by:  predtrss  6261  predpoirr  6272  predfrirr  6273  xpord2pred  34074
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