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Theorem elrefsymrelsrel 38572
Description: For sets, being an element of the class of reflexive and symmetric relations is equivalent to satisfying the reflexive and symmetric relation predicates. (Contributed by Peter Mazsa, 23-Aug-2021.)
Assertion
Ref Expression
elrefsymrelsrel (𝑅𝑉 → (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ( RefRel 𝑅 ∧ SymRel 𝑅)))

Proof of Theorem elrefsymrelsrel
StepHypRef Expression
1 elin 3967 . 2 (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ (𝑅 ∈ RefRels ∧ 𝑅 ∈ SymRels ))
2 elrefrelsrel 38521 . . 3 (𝑅𝑉 → (𝑅 ∈ RefRels ↔ RefRel 𝑅))
3 elsymrelsrel 38558 . . 3 (𝑅𝑉 → (𝑅 ∈ SymRels ↔ SymRel 𝑅))
42, 3anbi12d 632 . 2 (𝑅𝑉 → ((𝑅 ∈ RefRels ∧ 𝑅 ∈ SymRels ) ↔ ( RefRel 𝑅 ∧ SymRel 𝑅)))
51, 4bitrid 283 1 (𝑅𝑉 → (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ( RefRel 𝑅 ∧ SymRel 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2108  cin 3950   RefRels crefrels 38187   RefRel wrefrel 38188   SymRels csymrels 38193   SymRel wsymrel 38194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-rels 38486  df-ssr 38499  df-refs 38511  df-refrels 38512  df-refrel 38513  df-syms 38543  df-symrels 38544  df-symrel 38545
This theorem is referenced by: (None)
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