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Theorem eltrrelsrel 35922
 Description: For sets, being an element of the class of transitive relations is equivalent to satisfying the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.)
Assertion
Ref Expression
eltrrelsrel (𝑅𝑉 → (𝑅 ∈ TrRels ↔ TrRel 𝑅))

Proof of Theorem eltrrelsrel
StepHypRef Expression
1 elrelsrel 35832 . . 3 (𝑅𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
21anbi2d 631 . 2 (𝑅𝑉 → (((𝑅𝑅) ⊆ 𝑅𝑅 ∈ Rels ) ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅)))
3 eltrrels2 35920 . 2 (𝑅 ∈ TrRels ↔ ((𝑅𝑅) ⊆ 𝑅𝑅 ∈ Rels ))
4 dftrrel2 35918 . 2 ( TrRel 𝑅 ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅))
52, 3, 43bitr4g 317 1 (𝑅𝑉 → (𝑅 ∈ TrRels ↔ TrRel 𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2115   ⊆ wss 3919   ∘ ccom 5546  Rel wrel 5547   Rels crels 35560   TrRels ctrrels 35572   TrRel wtrrel 35573 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-rels 35830  df-ssr 35843  df-trs 35913  df-trrels 35914  df-trrel 35915 This theorem is referenced by: (None)
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