Theorem dfantisymrel4 39375

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 39374	. 2 ⊢ ( AntisymRel 𝑅 ↔ ( CnvRefRel (𝑅 ∩ ◡𝑅) ∧ Rel 𝑅))
2		relcnv 6097	. . . 4 ⊢ Rel ◡𝑅
3		relin2 5791	. . . 4 ⊢ (Rel ◡𝑅 → Rel (𝑅 ∩ ◡𝑅))
4	2, 3	ax-mp 5	. . 3 ⊢ Rel (𝑅 ∩ ◡𝑅)
5		dfcnvrefrel4 39123	. . 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 3906   ⊆ wss 3907   I cid 5546  ^◡ccnv 5651  Rel wrel 5657   CnvRefRel wcnvrefrel 38703   AntisymRel wantisymrel 38733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-cnv 5660  df-dm 5662  df-rn 5663  df-res 5664  df-cnvrefrel 39118  df-antisymrel 39374

This theorem is referenced by: (None)