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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 4577 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
2 | brun 3987 | . . . . 5 ⊢ (𝑥(𝑅 ∪ ◡𝑅)𝑦 ↔ (𝑥𝑅𝑦 ∨ 𝑥◡𝑅𝑦)) | |
3 | df-br 3938 | . . . . 5 ⊢ (𝑥(𝑅 ∪ ◡𝑅)𝑦 ↔ 〈𝑥, 𝑦〉 ∈ (𝑅 ∪ ◡𝑅)) | |
4 | vex 2692 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
5 | vex 2692 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
6 | 4, 5 | brcnv 4730 | . . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
7 | 6 | orbi2i 752 | . . . . 5 ⊢ ((𝑥𝑅𝑦 ∨ 𝑥◡𝑅𝑦) ↔ (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)) |
8 | 2, 3, 7 | 3bitr3i 209 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑅 ∪ ◡𝑅) ↔ (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)) |
9 | 1, 8 | imbi12i 238 | . . 3 ⊢ ((〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 〈𝑥, 𝑦〉 ∈ (𝑅 ∪ ◡𝑅)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))) |
10 | 9 | 2albii 1448 | . 2 ⊢ (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 〈𝑥, 𝑦〉 ∈ (𝑅 ∪ ◡𝑅)) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))) |
11 | relxp 4656 | . . 3 ⊢ Rel (𝐴 × 𝐵) | |
12 | ssrel 4635 | . . 3 ⊢ (Rel (𝐴 × 𝐵) → ((𝐴 × 𝐵) ⊆ (𝑅 ∪ ◡𝑅) ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 〈𝑥, 𝑦〉 ∈ (𝑅 ∪ ◡𝑅)))) | |
13 | 11, 12 | ax-mp 5 | . 2 ⊢ ((𝐴 × 𝐵) ⊆ (𝑅 ∪ ◡𝑅) ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 〈𝑥, 𝑦〉 ∈ (𝑅 ∪ ◡𝑅))) |
14 | r2al 2457 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))) | |
15 | 10, 13, 14 | 3bitr4i 211 | 1 ⊢ ((𝐴 × 𝐵) ⊆ (𝑅 ∪ ◡𝑅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 ∀wal 1330 ∈ wcel 1481 ∀wral 2417 ∪ cun 3074 ⊆ wss 3076 〈cop 3535 class class class wbr 3937 × cxp 4545 ◡ccnv 4546 Rel wrel 4552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-rel 4554 df-cnv 4555 |
This theorem is referenced by: (None) |
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