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Theorem eltrrelsrel 36681
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 36591 . . 3 (𝑅𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
21anbi2d 629 . 2 (𝑅𝑉 → (((𝑅𝑅) ⊆ 𝑅𝑅 ∈ Rels ) ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅)))
3 eltrrels2 36679 . 2 (𝑅 ∈ TrRels ↔ ((𝑅𝑅) ⊆ 𝑅𝑅 ∈ Rels ))
4 dftrrel2 36677 . 2 ( TrRel 𝑅 ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅))
52, 3, 43bitr4g 314 1 (𝑅𝑉 → (𝑅 ∈ TrRels ↔ TrRel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  wss 3887  ccom 5589  Rel wrel 5590   Rels crels 36321   TrRels ctrrels 36333   TrRel wtrrel 36334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pr 5351
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3432  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-br 5075  df-opab 5137  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-res 5597  df-rels 36589  df-ssr 36602  df-trs 36672  df-trrels 36673  df-trrel 36674
This theorem is referenced by: (None)
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