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Theorem predel 6139
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 4148 . 2 (𝑌 ∈ (𝐴 ∩ (𝑅 “ {𝑋})) → 𝑌𝐴)
2 df-pred 6122 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
31, 2eleq2s 2929 1 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  cin 3911  {csn 4541  ccnv 5528  cima 5532  Predcpred 6121
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-v 3475  df-in 3919  df-pred 6122
This theorem is referenced by:  predpo  6140  predpoirr  6150  predfrirr  6151  dftrpred3g  33077
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