dfantisymrel5 - Mathbox for Peter Mazsa

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Theorem dfantisymrel5 38744

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 38742	. 2 ⊢ ( AntisymRel 𝑅 ↔ ( CnvRefRel (𝑅 ∩ ^◡𝑅) ∧ Rel 𝑅))
2		relcnv 6125	. . . . 5 ⊢ Rel ^◡𝑅
3		relin2 5826	. . . . 5 ⊢ (Rel ^◡𝑅 → Rel (𝑅 ∩ ^◡𝑅))
4	2, 3	ax-mp 5	. . . 4 ⊢ Rel (𝑅 ∩ ^◡𝑅)
5		dfcnvrefrel5 38515	. . . 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 38352	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅 ∩ ^◡𝑅)𝑦 ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)))
8	7	el2v 3485	. . . . 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 1817	. . 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 1535 = wceq 1537 Vcvv 3478 ∩ cin 3962 class class class wbr 5148 ^◡ccnv 5688 Rel wrel 5694 CnvRefRel wcnvrefrel 38171 AntisymRel wantisymrel 38199

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-cnvrefrel 38509 df-antisymrel 38742

This theorem is referenced by: antisymrelres 38745 antisymrelressn 38746

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