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Mirrors > Home > ILE Home > Th. List > intasym | 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 4977 | . . 3 ⊢ Rel ◡𝑅 | |
2 | relin2 4718 | . . 3 ⊢ (Rel ◡𝑅 → Rel (𝑅 ∩ ◡𝑅)) | |
3 | ssrel 4687 | . . 3 ⊢ (Rel (𝑅 ∩ ◡𝑅) → ((𝑅 ∩ ◡𝑅) ⊆ I ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝑅 ∩ ◡𝑅) → 〈𝑥, 𝑦〉 ∈ I ))) | |
4 | 1, 2, 3 | mp2b 8 | . 2 ⊢ ((𝑅 ∩ ◡𝑅) ⊆ I ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝑅 ∩ ◡𝑅) → 〈𝑥, 𝑦〉 ∈ I )) |
5 | elin 3301 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑅 ∩ ◡𝑅) ↔ (〈𝑥, 𝑦〉 ∈ 𝑅 ∧ 〈𝑥, 𝑦〉 ∈ ◡𝑅)) | |
6 | df-br 3978 | . . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
7 | vex 2725 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
8 | vex 2725 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | brcnv 4782 | . . . . . . 7 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
10 | df-br 3978 | . . . . . . 7 ⊢ (𝑥◡𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ◡𝑅) | |
11 | 9, 10 | bitr3i 185 | . . . . . 6 ⊢ (𝑦𝑅𝑥 ↔ 〈𝑥, 𝑦〉 ∈ ◡𝑅) |
12 | 6, 11 | anbi12i 456 | . . . . 5 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (〈𝑥, 𝑦〉 ∈ 𝑅 ∧ 〈𝑥, 𝑦〉 ∈ ◡𝑅)) |
13 | 5, 12 | bitr4i 186 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑅 ∩ ◡𝑅) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)) |
14 | df-br 3978 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ I ) | |
15 | 8 | ideq 4751 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
16 | 14, 15 | bitr3i 185 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ I ↔ 𝑥 = 𝑦) |
17 | 13, 16 | imbi12i 238 | . . 3 ⊢ ((〈𝑥, 𝑦〉 ∈ (𝑅 ∩ ◡𝑅) → 〈𝑥, 𝑦〉 ∈ I ) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)) |
18 | 17 | 2albii 1458 | . 2 ⊢ (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝑅 ∩ ◡𝑅) → 〈𝑥, 𝑦〉 ∈ I ) ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)) |
19 | 4, 18 | bitri 183 | 1 ⊢ ((𝑅 ∩ ◡𝑅) ⊆ I ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1340 ∈ wcel 2135 ∩ cin 3111 ⊆ wss 3112 〈cop 3574 class class class wbr 3977 I cid 4261 ◡ccnv 4598 Rel wrel 4604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2724 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-br 3978 df-opab 4039 df-id 4266 df-xp 4605 df-rel 4606 df-cnv 4607 |
This theorem is referenced by: (None) |
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