dfantisymrel5 - Mathbox for Peter Mazsa

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Theorem dfantisymrel5 38881

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 38879	. 2 ⊢ ( AntisymRel 𝑅 ↔ ( CnvRefRel (𝑅 ∩ ^◡𝑅) ∧ Rel 𝑅))
2		relcnv 6057	. . . . 5 ⊢ Rel ^◡𝑅
3		relin2 5757	. . . . 5 ⊢ (Rel ^◡𝑅 → Rel (𝑅 ∩ ^◡𝑅))
4	2, 3	ax-mp 5	. . . 4 ⊢ Rel (𝑅 ∩ ^◡𝑅)
5		dfcnvrefrel5 38646	. . . 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 38423	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅 ∩ ^◡𝑅)𝑦 ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)))
8	7	el2v 3444	. . . . 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 1821	. . 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 1539 = wceq 1541 Vcvv 3437 ∩ cin 3897 class class class wbr 5093 ^◡ccnv 5618 Rel wrel 5624 CnvRefRel wcnvrefrel 38252 AntisymRel wantisymrel 38280

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pr 5372

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-sb 2068 df-clab 2712 df-cleq 2725 df-clel 2808 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4475 df-sn 4576 df-pr 4578 df-op 4582 df-br 5094 df-opab 5156 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-dm 5629 df-rn 5630 df-res 5631 df-cnvrefrel 38640 df-antisymrel 38879

This theorem is referenced by: antisymrelres 38882 antisymrelressn 38883

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