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Theorem predel 5858
 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 3942 . 2 (𝑌 ∈ (𝐴 ∩ (𝑅 “ {𝑋})) → 𝑌𝐴)
2 df-pred 5841 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
31, 2eleq2s 2857 1 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2139   ∩ cin 3714  {csn 4321  ◡ccnv 5265   “ cima 5269  Predcpred 5840 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722  df-pred 5841 This theorem is referenced by:  predpo  5859  predpoirr  5869  predfrirr  5870  dftrpred3g  32038
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