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Mirrors > Home > ILE Home > Th. List > qfto | GIF version |
Description: A quantifier-free way of expressing the total order predicate. (Contributed by Mario Carneiro, 22-Nov-2013.) |
Ref | Expression |
---|---|
qfto | ⊢ ((A × B) ⊆ (𝑅 ∪ ◡𝑅) ↔ ∀x ∈ A ∀y ∈ B (x𝑅y ∨ y𝑅x)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxp 4317 | . . . 4 ⊢ (〈x, y〉 ∈ (A × B) ↔ (x ∈ A ∧ y ∈ B)) | |
2 | brun 3801 | . . . . 5 ⊢ (x(𝑅 ∪ ◡𝑅)y ↔ (x𝑅y ∨ x◡𝑅y)) | |
3 | df-br 3756 | . . . . 5 ⊢ (x(𝑅 ∪ ◡𝑅)y ↔ 〈x, y〉 ∈ (𝑅 ∪ ◡𝑅)) | |
4 | vex 2554 | . . . . . . 7 ⊢ x ∈ V | |
5 | vex 2554 | . . . . . . 7 ⊢ y ∈ V | |
6 | 4, 5 | brcnv 4461 | . . . . . 6 ⊢ (x◡𝑅y ↔ y𝑅x) |
7 | 6 | orbi2i 678 | . . . . 5 ⊢ ((x𝑅y ∨ x◡𝑅y) ↔ (x𝑅y ∨ y𝑅x)) |
8 | 2, 3, 7 | 3bitr3i 199 | . . . 4 ⊢ (〈x, y〉 ∈ (𝑅 ∪ ◡𝑅) ↔ (x𝑅y ∨ y𝑅x)) |
9 | 1, 8 | imbi12i 228 | . . 3 ⊢ ((〈x, y〉 ∈ (A × B) → 〈x, y〉 ∈ (𝑅 ∪ ◡𝑅)) ↔ ((x ∈ A ∧ y ∈ B) → (x𝑅y ∨ y𝑅x))) |
10 | 9 | 2albii 1357 | . 2 ⊢ (∀x∀y(〈x, y〉 ∈ (A × B) → 〈x, y〉 ∈ (𝑅 ∪ ◡𝑅)) ↔ ∀x∀y((x ∈ A ∧ y ∈ B) → (x𝑅y ∨ y𝑅x))) |
11 | relxp 4390 | . . 3 ⊢ Rel (A × B) | |
12 | ssrel 4371 | . . 3 ⊢ (Rel (A × B) → ((A × B) ⊆ (𝑅 ∪ ◡𝑅) ↔ ∀x∀y(〈x, y〉 ∈ (A × B) → 〈x, y〉 ∈ (𝑅 ∪ ◡𝑅)))) | |
13 | 11, 12 | ax-mp 7 | . 2 ⊢ ((A × B) ⊆ (𝑅 ∪ ◡𝑅) ↔ ∀x∀y(〈x, y〉 ∈ (A × B) → 〈x, y〉 ∈ (𝑅 ∪ ◡𝑅))) |
14 | r2al 2337 | . 2 ⊢ (∀x ∈ A ∀y ∈ B (x𝑅y ∨ y𝑅x) ↔ ∀x∀y((x ∈ A ∧ y ∈ B) → (x𝑅y ∨ y𝑅x))) | |
15 | 10, 13, 14 | 3bitr4i 201 | 1 ⊢ ((A × B) ⊆ (𝑅 ∪ ◡𝑅) ↔ ∀x ∈ A ∀y ∈ B (x𝑅y ∨ y𝑅x)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∨ wo 628 ∀wal 1240 ∈ wcel 1390 ∀wral 2300 ∪ cun 2909 ⊆ wss 2911 〈cop 3370 class class class wbr 3755 × cxp 4286 ◡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-xp 4294 df-rel 4295 df-cnv 4296 |
This theorem is referenced by: (None) |
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