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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 6122 | . . 3 ⊢ Rel ◡𝑅 | |
| 2 | relin2 5823 | . . 3 ⊢ (Rel ◡𝑅 → Rel (𝑅 ∩ ◡𝑅)) | |
| 3 | ssrel 5792 | . . 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 3967 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑅 ∩ ◡𝑅) ↔ (〈𝑥, 𝑦〉 ∈ 𝑅 ∧ 〈𝑥, 𝑦〉 ∈ ◡𝑅)) | |
| 6 | df-br 5144 | . . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
| 7 | vex 3484 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
| 8 | vex 3484 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
| 9 | 7, 8 | brcnv 5893 | . . . . . . 7 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
| 10 | df-br 5144 | . . . . . . 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 5144 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ I ) | |
| 15 | 8 | ideq 5863 | . . . . 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 1820 | . 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 1538 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 〈cop 4632 class class class wbr 5143 I cid 5577 ◡ccnv 5684 Rel wrel 5690 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 |
| This theorem is referenced by: (None) |
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