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Mirrors > Home > MPE Home > Th. List > Mathboxes > antisymrelres | Structured version Visualization version GIF version |
Description: (Contributed by Peter Mazsa, 25-Jun-2024.) |
Ref | Expression |
---|---|
antisymrelres | ⊢ ( AntisymRel (𝑅 ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 6010 | . . 3 ⊢ Rel (𝑅 ↾ 𝐴) | |
2 | dfantisymrel5 37627 | . . 3 ⊢ ( AntisymRel (𝑅 ↾ 𝐴) ↔ (∀𝑥∀𝑦((𝑥(𝑅 ↾ 𝐴)𝑦 ∧ 𝑦(𝑅 ↾ 𝐴)𝑥) → 𝑥 = 𝑦) ∧ Rel (𝑅 ↾ 𝐴))) | |
3 | 1, 2 | mpbiran2 708 | . 2 ⊢ ( AntisymRel (𝑅 ↾ 𝐴) ↔ ∀𝑥∀𝑦((𝑥(𝑅 ↾ 𝐴)𝑦 ∧ 𝑦(𝑅 ↾ 𝐴)𝑥) → 𝑥 = 𝑦)) |
4 | brres 5988 | . . . . . . 7 ⊢ (𝑦 ∈ V → (𝑥(𝑅 ↾ 𝐴)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦))) | |
5 | 4 | elv 3480 | . . . . . 6 ⊢ (𝑥(𝑅 ↾ 𝐴)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)) |
6 | brres 5988 | . . . . . . 7 ⊢ (𝑥 ∈ V → (𝑦(𝑅 ↾ 𝐴)𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥))) | |
7 | 6 | elv 3480 | . . . . . 6 ⊢ (𝑦(𝑅 ↾ 𝐴)𝑥 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥)) |
8 | 5, 7 | anbi12i 627 | . . . . 5 ⊢ ((𝑥(𝑅 ↾ 𝐴)𝑦 ∧ 𝑦(𝑅 ↾ 𝐴)𝑥) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥))) |
9 | an4 654 | . . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))) | |
10 | 8, 9 | bitri 274 | . . . 4 ⊢ ((𝑥(𝑅 ↾ 𝐴)𝑦 ∧ 𝑦(𝑅 ↾ 𝐴)𝑥) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))) |
11 | 10 | imbi1i 349 | . . 3 ⊢ (((𝑥(𝑅 ↾ 𝐴)𝑦 ∧ 𝑦(𝑅 ↾ 𝐴)𝑥) → 𝑥 = 𝑦) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)) → 𝑥 = 𝑦)) |
12 | 11 | 2albii 1822 | . 2 ⊢ (∀𝑥∀𝑦((𝑥(𝑅 ↾ 𝐴)𝑦 ∧ 𝑦(𝑅 ↾ 𝐴)𝑥) → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)) → 𝑥 = 𝑦)) |
13 | r2alan 37111 | . 2 ⊢ (∀𝑥∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)) | |
14 | 3, 12, 13 | 3bitri 296 | 1 ⊢ ( AntisymRel (𝑅 ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 class class class wbr 5148 ↾ cres 5678 Rel wrel 5681 AntisymRel wantisymrel 37075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 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 37392 df-antisymrel 37625 |
This theorem is referenced by: (None) |
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