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Theorem eltrrelsrel 37072
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 36978 . . 3 (𝑅𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
21anbi2d 630 . 2 (𝑅𝑉 → (((𝑅𝑅) ⊆ 𝑅𝑅 ∈ Rels ) ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅)))
3 eltrrels2 37070 . 2 (𝑅 ∈ TrRels ↔ ((𝑅𝑅) ⊆ 𝑅𝑅 ∈ Rels ))
4 dftrrel2 37068 . 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 397  wcel 2107  wss 3915  ccom 5642  Rel wrel 5643   Rels crels 36665   TrRels ctrrels 36677   TrRel wtrrel 36678
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-rels 36976  df-ssr 36989  df-trs 37063  df-trrels 37064  df-trrel 37065
This theorem is referenced by: (None)
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