Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elrefsymrelsrel Structured version   Visualization version   GIF version

Theorem elrefsymrelsrel 35960
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 3900 . 2 (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ (𝑅 ∈ RefRels ∧ 𝑅 ∈ SymRels ))
2 elrefrelsrel 35912 . . 3 (𝑅𝑉 → (𝑅 ∈ RefRels ↔ RefRel 𝑅))
3 elsymrelsrel 35946 . . 3 (𝑅𝑉 → (𝑅 ∈ SymRels ↔ SymRel 𝑅))
42, 3anbi12d 633 . 2 (𝑅𝑉 → ((𝑅 ∈ RefRels ∧ 𝑅 ∈ SymRels ) ↔ ( RefRel 𝑅 ∧ SymRel 𝑅)))
51, 4syl5bb 286 1 (𝑅𝑉 → (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ( RefRel 𝑅 ∧ SymRel 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2112  cin 3883   RefRels crefrels 35611   RefRel wrefrel 35612   SymRels csymrels 35617   SymRel wsymrel 35618
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-rels 35878  df-ssr 35891  df-refs 35903  df-refrels 35904  df-refrel 35905  df-syms 35931  df-symrels 35932  df-symrel 35933
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator