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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfantisymrel5 | Structured version Visualization version GIF version |
Description: Alternate definition of the antisymmetric relation predicate. (Contributed by Peter Mazsa, 24-Jun-2024.) |
Ref | Expression |
---|---|
dfantisymrel5 | ⊢ ( AntisymRel 𝑅 ↔ (∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ Rel 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-antisymrel 37619 | . 2 ⊢ ( AntisymRel 𝑅 ↔ ( CnvRefRel (𝑅 ∩ ◡𝑅) ∧ Rel 𝑅)) | |
2 | relcnv 6101 | . . . . 5 ⊢ Rel ◡𝑅 | |
3 | relin2 5812 | . . . . 5 ⊢ (Rel ◡𝑅 → Rel (𝑅 ∩ ◡𝑅)) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ Rel (𝑅 ∩ ◡𝑅) |
5 | dfcnvrefrel5 37392 | . . . 4 ⊢ ( CnvRefRel (𝑅 ∩ ◡𝑅) ↔ (∀𝑥∀𝑦(𝑥(𝑅 ∩ ◡𝑅)𝑦 → 𝑥 = 𝑦) ∧ Rel (𝑅 ∩ ◡𝑅))) | |
6 | 4, 5 | mpbiran2 709 | . . 3 ⊢ ( CnvRefRel (𝑅 ∩ ◡𝑅) ↔ ∀𝑥∀𝑦(𝑥(𝑅 ∩ ◡𝑅)𝑦 → 𝑥 = 𝑦)) |
7 | brcnvin 37229 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅 ∩ ◡𝑅)𝑦 ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))) | |
8 | 7 | el2v 3483 | . . . . 5 ⊢ (𝑥(𝑅 ∩ ◡𝑅)𝑦 ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)) |
9 | 8 | imbi1i 350 | . . . 4 ⊢ ((𝑥(𝑅 ∩ ◡𝑅)𝑦 → 𝑥 = 𝑦) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)) |
10 | 9 | 2albii 1823 | . . 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 37084 | 1 ⊢ ( AntisymRel 𝑅 ↔ (∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ Rel 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 = wceq 1542 Vcvv 3475 ∩ cin 3947 class class class wbr 5148 ◡ccnv 5675 Rel wrel 5681 CnvRefRel wcnvrefrel 37041 AntisymRel wantisymrel 37069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-cnvrefrel 37386 df-antisymrel 37619 |
This theorem is referenced by: antisymrelres 37622 antisymrelressn 37623 |
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