Theorem dfantisymrel4 39402

Description: Alternate definition of the antisymmetric relation predicate. (Contributed by Peter Mazsa, 24-Jun-2024.)

Assertion

Ref	Expression
dfantisymrel4	⊢ ( AntisymRel 𝑅 ↔ ((𝑅 ∩ ◡𝑅) ⊆ I ∧ Rel 𝑅))

Proof of Theorem dfantisymrel4

Step	Hyp	Ref	Expression
1		df-antisymrel 39401	. 2 ⊢ ( AntisymRel 𝑅 ↔ ( CnvRefRel (𝑅 ∩ ◡𝑅) ∧ Rel 𝑅))
2		relcnv 6107	. . . 4 ⊢ Rel ◡𝑅
3		relin2 5801	. . . 4 ⊢ (Rel ◡𝑅 → Rel (𝑅 ∩ ◡𝑅))
4	2, 3	ax-mp 5	. . 3 ⊢ Rel (𝑅 ∩ ◡𝑅)
5		dfcnvrefrel4 39150	. . 3 ⊢ ( CnvRefRel (𝑅 ∩ ◡𝑅) ↔ ((𝑅 ∩ ◡𝑅) ⊆ I ∧ Rel (𝑅 ∩ ◡𝑅)))
6	4, 5	mpbiran2 722	. 2 ⊢ ( CnvRefRel (𝑅 ∩ ◡𝑅) ↔ (𝑅 ∩ ◡𝑅) ⊆ I )
7	1, 6	bianbi 638	1 ⊢ ( AntisymRel 𝑅 ↔ ((𝑅 ∩ ◡𝑅) ⊆ I ∧ Rel 𝑅))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∩ cin 3912   ⊆ wss 3913   I cid 5556  ^◡ccnv 5661  Rel wrel 5667   CnvRefRel wcnvrefrel 38730   AntisymRel wantisymrel 38760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-cnvrefrel 39145  df-antisymrel 39401

This theorem is referenced by: (None)