Theorem dfantisymrel4 39494

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 39493	. 2 ⊢ ( AntisymRel 𝑅 ↔ ( CnvRefRel (𝑅 ∩ ◡𝑅) ∧ Rel 𝑅))
2		relcnv 6108	. . . 4 ⊢ Rel ◡𝑅
3		relin2 5802	. . . 4 ⊢ (Rel ◡𝑅 → Rel (𝑅 ∩ ◡𝑅))
4	2, 3	ax-mp 5	. . 3 ⊢ Rel (𝑅 ∩ ◡𝑅)
5		dfcnvrefrel4 39242	. . 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 3905   ⊆ wss 3906   I cid 5557  ^◡ccnv 5662  Rel wrel 5668   CnvRefRel wcnvrefrel 38822   AntisymRel wantisymrel 38852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-cnvrefrel 39237  df-antisymrel 39493

This theorem is referenced by: (None)