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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 5018 | . . 3 ⊢ Rel ◡𝑅 | |
2 | relin2 4757 | . . 3 ⊢ (Rel ◡𝑅 → Rel (𝑅 ∩ ◡𝑅)) | |
3 | ssrel 4726 | . . 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 3330 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑅 ∩ ◡𝑅) ↔ (〈𝑥, 𝑦〉 ∈ 𝑅 ∧ 〈𝑥, 𝑦〉 ∈ ◡𝑅)) | |
6 | df-br 4016 | . . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
7 | vex 2752 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
8 | vex 2752 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | brcnv 4822 | . . . . . . 7 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
10 | df-br 4016 | . . . . . . 7 ⊢ (𝑥◡𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ◡𝑅) | |
11 | 9, 10 | bitr3i 186 | . . . . . 6 ⊢ (𝑦𝑅𝑥 ↔ 〈𝑥, 𝑦〉 ∈ ◡𝑅) |
12 | 6, 11 | anbi12i 460 | . . . . 5 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (〈𝑥, 𝑦〉 ∈ 𝑅 ∧ 〈𝑥, 𝑦〉 ∈ ◡𝑅)) |
13 | 5, 12 | bitr4i 187 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑅 ∩ ◡𝑅) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥)) |
14 | df-br 4016 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ I ) | |
15 | 8 | ideq 4791 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
16 | 14, 15 | bitr3i 186 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ I ↔ 𝑥 = 𝑦) |
17 | 13, 16 | imbi12i 239 | . . 3 ⊢ ((〈𝑥, 𝑦〉 ∈ (𝑅 ∩ ◡𝑅) → 〈𝑥, 𝑦〉 ∈ I ) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)) |
18 | 17 | 2albii 1481 | . 2 ⊢ (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝑅 ∩ ◡𝑅) → 〈𝑥, 𝑦〉 ∈ I ) ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)) |
19 | 4, 18 | bitri 184 | 1 ⊢ ((𝑅 ∩ ◡𝑅) ⊆ I ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∀wal 1361 ∈ wcel 2158 ∩ cin 3140 ⊆ wss 3141 〈cop 3607 class class class wbr 4015 I cid 4300 ◡ccnv 4637 Rel wrel 4643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 |
This theorem is referenced by: (None) |
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