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Theorem predss 5604
 Description: The predecessor class of 𝐴 is a subset of 𝐴. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
predss Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴

Proof of Theorem predss
StepHypRef Expression
1 df-pred 5597 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
2 inss1 3795 . 2 (𝐴 ∩ (𝑅 “ {𝑋})) ⊆ 𝐴
31, 2eqsstri 3598 1 Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∩ cin 3539   ⊆ wss 3540  {csn 4125  ◡ccnv 5037   “ cima 5041  Predcpred 5596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-pred 5597 This theorem is referenced by:  wfr3g  7300  wfrlem4  7305  wfrlem10  7311  trpredlem1  30971  wsuclem  31017  wsuclemOLD  31018  frr3g  31023  frrlem4  31027
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