dfantisymrel5 - Mathbox for Peter Mazsa

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Theorem dfantisymrel5 39328

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 39326	. 2 ⊢ ( AntisymRel 𝑅 ↔ ( CnvRefRel (𝑅 ∩ ^◡𝑅) ∧ Rel 𝑅))
2		relcnv 6090	. . . . 5 ⊢ Rel ^◡𝑅
3		relin2 5784	. . . . 5 ⊢ (Rel ^◡𝑅 → Rel (𝑅 ∩ ^◡𝑅))
4	2, 3	ax-mp 5	. . . 4 ⊢ Rel (𝑅 ∩ ^◡𝑅)
5		dfcnvrefrel5 39076	. . . 4 ⊢ ( CnvRefRel (𝑅 ∩ ^◡𝑅) ↔ (∀𝑥∀𝑦(𝑥(𝑅 ∩ ^◡𝑅)𝑦 → 𝑥 = 𝑦) ∧ Rel (𝑅 ∩ ^◡𝑅)))
6	4, 5	mpbiran2 720	. . 3 ⊢ ( CnvRefRel (𝑅 ∩ ^◡𝑅) ↔ ∀𝑥∀𝑦(𝑥(𝑅 ∩ ^◡𝑅)𝑦 → 𝑥 = 𝑦))
7		brcnvin 38841	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅 ∩ ^◡𝑅)𝑦 ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)))
8	7	el2v 3460	. . . . 5 ⊢ (𝑥(𝑅 ∩ ^◡𝑅)𝑦 ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))
9	8	imbi1i 351	. . . 4 ⊢ ((𝑥(𝑅 ∩ ^◡𝑅)𝑦 → 𝑥 = 𝑦) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
10	9	2albii 1839	. . 3 ⊢ (∀𝑥∀𝑦(𝑥(𝑅 ∩ ^◡𝑅)𝑦 → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
11	6, 10	bitri 277	. 2 ⊢ ( CnvRefRel (𝑅 ∩ ^◡𝑅) ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
12	1, 11	bianbi 636	1 ⊢ ( AntisymRel 𝑅 ↔ (∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ Rel 𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∀wal 1557 = wceq 1559 Vcvv 3453 ∩ cin 3903 class class class wbr 5099 ^◡ccnv 5644 Rel wrel 5650 CnvRefRel wcnvrefrel 38655 AntisymRel wantisymrel 38685

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-ext 2733 ax-sep 5245 ax-pr 5389

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-sb 2090 df-clab 2740 df-cleq 2753 df-clel 2836 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4480 df-sn 4582 df-pr 4584 df-op 4588 df-br 5100 df-opab 5162 df-id 5540 df-xp 5651 df-rel 5652 df-cnv 5653 df-dm 5655 df-rn 5656 df-res 5657 df-cnvrefrel 39070 df-antisymrel 39326

This theorem is referenced by: antisymrelres 39329 antisymrelressn 39330

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