dfantisymrel5 - Mathbox for Peter Mazsa

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Theorem dfantisymrel5 38763

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

**Assertion**
Ref	Expression
dfantisymrel5	⊢ ( AntisymRel 𝑅 ↔ (∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ Rel 𝑅))

Distinct variable group: 𝑥,𝑅,𝑦

**Proof of Theorem dfantisymrel5**
Step	Hyp	Ref	Expression
1		df-antisymrel 38761	. 2 ⊢ ( AntisymRel 𝑅 ↔ ( CnvRefRel (𝑅 ∩ ^◡𝑅) ∧ Rel 𝑅))
2		relcnv 6122	. . . . 5 ⊢ Rel ^◡𝑅
3		relin2 5823	. . . . 5 ⊢ (Rel ^◡𝑅 → Rel (𝑅 ∩ ^◡𝑅))
4	2, 3	ax-mp 5	. . . 4 ⊢ Rel (𝑅 ∩ ^◡𝑅)
5		dfcnvrefrel5 38534	. . . 4 ⊢ ( CnvRefRel (𝑅 ∩ ^◡𝑅) ↔ (∀𝑥∀𝑦(𝑥(𝑅 ∩ ^◡𝑅)𝑦 → 𝑥 = 𝑦) ∧ Rel (𝑅 ∩ ^◡𝑅)))
6	4, 5	mpbiran2 710	. . 3 ⊢ ( CnvRefRel (𝑅 ∩ ^◡𝑅) ↔ ∀𝑥∀𝑦(𝑥(𝑅 ∩ ^◡𝑅)𝑦 → 𝑥 = 𝑦))
7		brcnvin 38371	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅 ∩ ^◡𝑅)𝑦 ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)))
8	7	el2v 3487	. . . . 5 ⊢ (𝑥(𝑅 ∩ ^◡𝑅)𝑦 ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))
9	8	imbi1i 349	. . . 4 ⊢ ((𝑥(𝑅 ∩ ^◡𝑅)𝑦 → 𝑥 = 𝑦) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
10	9	2albii 1820	. . 3 ⊢ (∀𝑥∀𝑦(𝑥(𝑅 ∩ ^◡𝑅)𝑦 → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
11	6, 10	bitri 275	. 2 ⊢ ( CnvRefRel (𝑅 ∩ ^◡𝑅) ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
12	1, 11	bianbi 627	1 ⊢ ( AntisymRel 𝑅 ↔ (∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ Rel 𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 = wceq 1540 Vcvv 3480 ∩ cin 3950 class class class wbr 5143 ^◡ccnv 5684 Rel wrel 5690 CnvRefRel wcnvrefrel 38191 AntisymRel wantisymrel 38219

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-cnvrefrel 38528 df-antisymrel 38761

This theorem is referenced by: antisymrelres 38764 antisymrelressn 38765

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