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Theorem dfantisymrel5 37621
Description: Alternate definition of the antisymmetric relation predicate. (Contributed by Peter Mazsa, 24-Jun-2024.)
Assertion
Ref Expression
dfantisymrel5 ( AntisymRel 𝑅 ↔ (∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ Rel 𝑅))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem dfantisymrel5
StepHypRef Expression
1 df-antisymrel 37619 . 2 ( AntisymRel 𝑅 ↔ ( CnvRefRel (𝑅𝑅) ∧ Rel 𝑅))
2 relcnv 6101 . . . . 5 Rel 𝑅
3 relin2 5812 . . . . 5 (Rel 𝑅 → Rel (𝑅𝑅))
42, 3ax-mp 5 . . . 4 Rel (𝑅𝑅)
5 dfcnvrefrel5 37392 . . . 4 ( CnvRefRel (𝑅𝑅) ↔ (∀𝑥𝑦(𝑥(𝑅𝑅)𝑦𝑥 = 𝑦) ∧ Rel (𝑅𝑅)))
64, 5mpbiran2 709 . . 3 ( CnvRefRel (𝑅𝑅) ↔ ∀𝑥𝑦(𝑥(𝑅𝑅)𝑦𝑥 = 𝑦))
7 brcnvin 37229 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅𝑅)𝑦 ↔ (𝑥𝑅𝑦𝑦𝑅𝑥)))
87el2v 3483 . . . . 5 (𝑥(𝑅𝑅)𝑦 ↔ (𝑥𝑅𝑦𝑦𝑅𝑥))
98imbi1i 350 . . . 4 ((𝑥(𝑅𝑅)𝑦𝑥 = 𝑦) ↔ ((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
1092albii 1823 . . 3 (∀𝑥𝑦(𝑥(𝑅𝑅)𝑦𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
116, 10bitri 275 . 2 ( CnvRefRel (𝑅𝑅) ↔ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
121, 11bianbi 37084 1 ( AntisymRel 𝑅 ↔ (∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ Rel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  Vcvv 3475  cin 3947   class class class wbr 5148  ccnv 5675  Rel wrel 5681   CnvRefRel wcnvrefrel 37041   AntisymRel wantisymrel 37069
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 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427
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 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-cnvrefrel 37386  df-antisymrel 37619
This theorem is referenced by:  antisymrelres  37622  antisymrelressn  37623
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