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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 4823 | . . 3 ⊢ Rel ◡𝑅 | |
2 | relin2 4569 | . . 3 ⊢ (Rel ◡𝑅 → Rel (𝑅 ∩ ◡𝑅)) | |
3 | ssrel 4539 | . . 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 3184 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑅 ∩ ◡𝑅) ↔ (〈𝑥, 𝑦〉 ∈ 𝑅 ∧ 〈𝑥, 𝑦〉 ∈ ◡𝑅)) | |
6 | df-br 3852 | . . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
7 | vex 2623 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
8 | vex 2623 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | brcnv 4632 | . . . . . . 7 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
10 | df-br 3852 | . . . . . . 7 ⊢ (𝑥◡𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ◡𝑅) | |
11 | 9, 10 | bitr3i 185 | . . . . . 6 ⊢ (𝑦𝑅𝑥 ↔ 〈𝑥, 𝑦〉 ∈ ◡𝑅) |
12 | 6, 11 | anbi12i 449 | . . . . 5 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (〈𝑥, 𝑦〉 ∈ 𝑅 ∧ 〈𝑥, 𝑦〉 ∈ ◡𝑅)) |
13 | 5, 12 | bitr4i 186 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑅 ∩ ◡𝑅) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)) |
14 | df-br 3852 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ I ) | |
15 | 8 | ideq 4601 | . . . . 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 1406 | . 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 1288 ∈ wcel 1439 ∩ cin 2999 ⊆ wss 3000 〈cop 3453 class class class wbr 3851 I cid 4124 ◡ccnv 4451 Rel wrel 4457 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-br 3852 df-opab 3906 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 |
This theorem is referenced by: (None) |
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