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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 4646 | . . 3 ⊢ Rel ◡𝑅 | |
2 | relin2 4399 | . . 3 ⊢ (Rel ◡𝑅 → Rel (𝑅 ∩ ◡𝑅)) | |
3 | ssrel 4371 | . . 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 3120 | . . . . 5 ⊢ (〈x, y〉 ∈ (𝑅 ∩ ◡𝑅) ↔ (〈x, y〉 ∈ 𝑅 ∧ 〈x, y〉 ∈ ◡𝑅)) | |
6 | df-br 3756 | . . . . . 6 ⊢ (x𝑅y ↔ 〈x, y〉 ∈ 𝑅) | |
7 | vex 2554 | . . . . . . . 8 ⊢ x ∈ V | |
8 | vex 2554 | . . . . . . . 8 ⊢ y ∈ V | |
9 | 7, 8 | brcnv 4461 | . . . . . . 7 ⊢ (x◡𝑅y ↔ y𝑅x) |
10 | df-br 3756 | . . . . . . 7 ⊢ (x◡𝑅y ↔ 〈x, y〉 ∈ ◡𝑅) | |
11 | 9, 10 | bitr3i 175 | . . . . . 6 ⊢ (y𝑅x ↔ 〈x, y〉 ∈ ◡𝑅) |
12 | 6, 11 | anbi12i 433 | . . . . 5 ⊢ ((x𝑅y ∧ y𝑅x) ↔ (〈x, y〉 ∈ 𝑅 ∧ 〈x, y〉 ∈ ◡𝑅)) |
13 | 5, 12 | bitr4i 176 | . . . 4 ⊢ (〈x, y〉 ∈ (𝑅 ∩ ◡𝑅) ↔ (x𝑅y ∧ y𝑅x)) |
14 | df-br 3756 | . . . . 5 ⊢ (x I y ↔ 〈x, y〉 ∈ I ) | |
15 | 8 | ideq 4431 | . . . . 5 ⊢ (x I y ↔ x = y) |
16 | 14, 15 | bitr3i 175 | . . . 4 ⊢ (〈x, y〉 ∈ I ↔ x = y) |
17 | 13, 16 | imbi12i 228 | . . 3 ⊢ ((〈x, y〉 ∈ (𝑅 ∩ ◡𝑅) → 〈x, y〉 ∈ I ) ↔ ((x𝑅y ∧ y𝑅x) → x = y)) |
18 | 17 | 2albii 1357 | . 2 ⊢ (∀x∀y(〈x, y〉 ∈ (𝑅 ∩ ◡𝑅) → 〈x, y〉 ∈ I ) ↔ ∀x∀y((x𝑅y ∧ y𝑅x) → x = y)) |
19 | 4, 18 | bitri 173 | 1 ⊢ ((𝑅 ∩ ◡𝑅) ⊆ I ↔ ∀x∀y((x𝑅y ∧ y𝑅x) → x = y)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1240 ∈ wcel 1390 ∩ cin 2910 ⊆ wss 2911 〈cop 3370 class class class wbr 3755 I cid 4016 ◡ccnv 4287 Rel wrel 4293 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 |
This theorem is referenced by: (None) |
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