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Mirrors > Home > MPE Home > Th. List > qfto | Structured version Visualization version 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 5486 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
2 | brun 5019 | . . . . 5 ⊢ (𝑥(𝑅 ∪ ◡𝑅)𝑦 ↔ (𝑥𝑅𝑦 ∨ 𝑥◡𝑅𝑦)) | |
3 | df-br 4969 | . . . . 5 ⊢ (𝑥(𝑅 ∪ ◡𝑅)𝑦 ↔ 〈𝑥, 𝑦〉 ∈ (𝑅 ∪ ◡𝑅)) | |
4 | vex 3443 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
5 | vex 3443 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
6 | 4, 5 | brcnv 5646 | . . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
7 | 6 | orbi2i 907 | . . . . 5 ⊢ ((𝑥𝑅𝑦 ∨ 𝑥◡𝑅𝑦) ↔ (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)) |
8 | 2, 3, 7 | 3bitr3i 302 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑅 ∪ ◡𝑅) ↔ (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)) |
9 | 1, 8 | imbi12i 352 | . . 3 ⊢ ((〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 〈𝑥, 𝑦〉 ∈ (𝑅 ∪ ◡𝑅)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))) |
10 | 9 | 2albii 1806 | . 2 ⊢ (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 〈𝑥, 𝑦〉 ∈ (𝑅 ∪ ◡𝑅)) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))) |
11 | relxp 5468 | . . 3 ⊢ Rel (𝐴 × 𝐵) | |
12 | ssrel 5550 | . . 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 3170 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))) | |
15 | 10, 13, 14 | 3bitr4i 304 | 1 ⊢ ((𝐴 × 𝐵) ⊆ (𝑅 ∪ ◡𝑅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 842 ∀wal 1523 ∈ wcel 2083 ∀wral 3107 ∪ cun 3863 ⊆ wss 3865 〈cop 4484 class class class wbr 4968 × cxp 5448 ◡ccnv 5449 Rel wrel 5455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pr 5228 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-br 4969 df-opab 5031 df-xp 5456 df-rel 5457 df-cnv 5458 |
This theorem is referenced by: istsr2 17661 letsr 17670 |
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