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Theorem predpredss 5603
 Description: If 𝐴 is a subset of 𝐵, then their predecessor classes are also subsets. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
predpredss (𝐴𝐵 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, 𝐵, 𝑋))

Proof of Theorem predpredss
StepHypRef Expression
1 ssrin 3800 . 2 (𝐴𝐵 → (𝐴 ∩ (𝑅 “ {𝑋})) ⊆ (𝐵 ∩ (𝑅 “ {𝑋})))
2 df-pred 5597 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
3 df-pred 5597 . 2 Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (𝑅 “ {𝑋}))
41, 2, 33sstr4g 3609 1 (𝐴𝐵 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, 𝐵, 𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∩ 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:  preddowncl  5624  wfrlem8  7309
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