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Mirrors > Home > MPE Home > Th. List > intasym | Structured version Visualization version GIF version |
Description: Two ways of saying a relation is antisymmetric. Definition of antisymmetry in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
intasym | ⊢ ((𝑅 ∩ ◡𝑅) ⊆ I ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 6124 | . . 3 ⊢ Rel ◡𝑅 | |
2 | relin2 5825 | . . 3 ⊢ (Rel ◡𝑅 → Rel (𝑅 ∩ ◡𝑅)) | |
3 | ssrel 5794 | . . 3 ⊢ (Rel (𝑅 ∩ ◡𝑅) → ((𝑅 ∩ ◡𝑅) ⊆ I ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝑅 ∩ ◡𝑅) → 〈𝑥, 𝑦〉 ∈ I ))) | |
4 | 1, 2, 3 | mp2b 10 | . 2 ⊢ ((𝑅 ∩ ◡𝑅) ⊆ I ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝑅 ∩ ◡𝑅) → 〈𝑥, 𝑦〉 ∈ I )) |
5 | elin 3978 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑅 ∩ ◡𝑅) ↔ (〈𝑥, 𝑦〉 ∈ 𝑅 ∧ 〈𝑥, 𝑦〉 ∈ ◡𝑅)) | |
6 | df-br 5148 | . . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
7 | vex 3481 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
8 | vex 3481 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | brcnv 5895 | . . . . . . 7 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
10 | df-br 5148 | . . . . . . 7 ⊢ (𝑥◡𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ◡𝑅) | |
11 | 9, 10 | bitr3i 277 | . . . . . 6 ⊢ (𝑦𝑅𝑥 ↔ 〈𝑥, 𝑦〉 ∈ ◡𝑅) |
12 | 6, 11 | anbi12i 628 | . . . . 5 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (〈𝑥, 𝑦〉 ∈ 𝑅 ∧ 〈𝑥, 𝑦〉 ∈ ◡𝑅)) |
13 | 5, 12 | bitr4i 278 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑅 ∩ ◡𝑅) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)) |
14 | df-br 5148 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ I ) | |
15 | 8 | ideq 5865 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
16 | 14, 15 | bitr3i 277 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ I ↔ 𝑥 = 𝑦) |
17 | 13, 16 | imbi12i 350 | . . 3 ⊢ ((〈𝑥, 𝑦〉 ∈ (𝑅 ∩ ◡𝑅) → 〈𝑥, 𝑦〉 ∈ I ) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)) |
18 | 17 | 2albii 1816 | . 2 ⊢ (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝑅 ∩ ◡𝑅) → 〈𝑥, 𝑦〉 ∈ I ) ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)) |
19 | 4, 18 | bitri 275 | 1 ⊢ ((𝑅 ∩ ◡𝑅) ⊆ I ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1534 ∈ wcel 2105 ∩ cin 3961 ⊆ wss 3962 〈cop 4636 class class class wbr 5147 I cid 5581 ◡ccnv 5687 Rel wrel 5693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 |
This theorem is referenced by: (None) |
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